Logarithms Explained Rules & Properties, Condense, Expand, Graphing & Solving Equations Introduction

This algebra 2 / precalculus math video tutorial explains the rules and properties of logarithms. It shows you how to condense and expand a logarithmic expression in addition to graphing and solving logarithmic equations.
Logarithms - The Easy Way!
• Logarithms - The Easy ...
Log to Exponential Form:
• Writing Logarithmic Eq...
Change of Base Formula:
• Change of Base Formula...
Change of Base Log Problem:
• Logarithms
Properties of Logarithms:
• Properties of Logarithms
____________________________________
Expanding Logarithmic Expressions:
• Expanding Logarithmic ...
Condensing Logarithmic Expressions:
• Condensing Logarithmic...
Natural Logarithms:
• Natural Logarithms
Solving Exponential Equations:
• Solving Exponential Eq...
Exponential Equations - Quadratic Form:
• Solving Exponential Eq...
_______________________________________
Solving Logarithmic Equations:
• Solving Logarithmic Eq...
Graphing Logarithmic Functions:
• Graphing Logarithmic F...
Graphing Exponential Functions:
• How To Graph Exponenti...
Compound Interest Word Problems:
• Compound Interest
Logarithms Practice Problems:
• Logarithms - Practice ...
_______________________________________
Final Exams and Video Playlists:
www.video-tutor.net/
Full-Length Videos and Worksheets:
/ collections

[@TheOrganicChemistryTutor]

Final Exams and Video Playlists: www.video-tutor.net/ Full-Length Videos & Worksheets: www.patreon.com/MathScienceTutor/collections

[@BerylliumOxide]

After learning all this stuff about logs, I'm ready to become a lumberjack.

[@workoutchannel4411]

underrated comment

[@bigboichan5992]

I SAW this one coming

[@BJamesThompson]

Natural "e" 😏

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What is the lumberjeck

[@treasuretotrash2067]

Your a carcinogen, literally. BeO is toxic

[@archeribull9869]

Not the hero we deserved, but the hero we need.

[@biswaranjan1298]

yes...

[@figg7366]

500th like..nice

[@spicedcoffee9902]

666 likes

[@SandhyaSharma-wr1vv]

true

[@securedthing6970]

1k likes

[@anyavaldez281]

In my opinion, the best online tutorial ever for logarithms

[@the44thchamber]

I agree. No one has presented the topic this thoroughly. PatrickJMT and Organic Chem are the two best virtual tutors imo, but organic chem far exceeded Patrick in this review.

[@jamesbetti9002]

Edward Breault Sal have anything on this?

[@sr-cg9li]

Nope. Khan Academy Is Obviously The Best.

[@gislentim8425]

really the best

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damnnnnnn, you fine

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I'm gonna give you a shout out when i graduate

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did you graduate

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🎯 Key Takeaways for quick navigation: 00:27 🧠 *When evaluating logarithms, ask yourself what power the base must be raised to get the given number.* 02:21 📏 *Logarithms can be evaluated by counting the number of times a base must be multiplied to reach a given number.* 04:03 🔢 *The logarithm of 10 is 1, log of 100 is 2, log of 1000 is 3, and so on. Counting zeros in the number gives the logarithm base 10.* 05:49 ⚠️ *Logarithm of 0 and logarithm of a negative number are undefined.* 08:32 📐 *The change of base formula: log base a of b can be expressed as log base c of b divided by log base c of a.* 12:45 📈 *Properties: log(a) + log(b) = log(a * b), log(a) - log(b) = log(a / b), log(a^2) = 2 * log(a).* 14:23 ➗ *When condensing log expressions with addition or subtraction, positive terms go on top, negative terms on the bottom.* 16:36 🔄 *When condensing log expressions with coefficients, use the power rule and move the coefficients to the exponents.* 20:36 🌐 *Expanding log expressions involves distributing the exponents and simplifying the result.* 22:22 🔄 *Simplifying expressions involving natural logarithms often involves canceling out the ln and e terms.* 24:50 🧩 *When solving logarithmic equations, converting to exponential form helps find the unknown variable.* 28:05 🔄 *Exponential form to logarithmic form conversion involves setting the exponent as the log result.* 30:48 ➡️ *Solving logarithmic equations involves converting them to exponential form and solving for the variable.* 31:57 🧮 *When solving equations with logarithms, apply inverse operations, such as subtraction or division, to isolate the variable.* 33:28 📊 *When dealing with multiple logarithms, condense them into a single logarithm using multiplication to simplify problem-solving.* 36:54 🚫 *Be cautious of extraneous solutions when solving logarithmic equations, as they might lead to invalid results.* 37:26 📐 *When combining logarithmic expressions, use addition for expressions with different bases, and multiply for expressions with the same base.* 48:11 🧠 *When faced with an exponential equation with different bases, try to express them in a common base for easier comparison and solution.* 54:28 📈 *Determine the domain of a logarithmic function by ensuring that the argument inside the logarithm is greater than zero.* 01:00:13 ↔️ *To find the inverse function of a logarithmic function, switch x and y, solve for y, and replace y with the inverse function symbol.* 01:01:46 📊 *Graphs of exponential functions and logarithmic functions are inversely related, demonstrating the fundamental connection between them.* 01:02:42 📈 *Exponential functions have horizontal asymptotes, and the equation for the horizontal asymptote can be found using the external number in the exponential function. For example, if it's 2^x + 1, the horizontal asymptote is y = 1.* 01:03:11 📉 *Logarithmic functions have vertical asymptotes, and the equation for the vertical asymptote can be found by setting the inside of the logarithm equal to 0. For example, log base 2 (x - 3) implies a vertical asymptote at x = 3.* 01:03:24 📊 *Once you have asymptotes and two points, you can graph an exponential function. Start by plotting the asymptotes and then follow the points to graph the curve.* 01:05:42 🔄 *The horizontal asymptote for an exponential function is the number outside the exponential part. For example, in the function e^(x - 1) - 2, the horizontal asymptote is y = -2.* 01:08:05 📉 *For functions like 2^(4 - x), find the points by setting the exponent equal to 0 and 1. Plug these values into the equation to get points for graphing. The horizontal asymptote is the number outside the exponential part.* 01:11:21 📊 *To graph logarithmic functions like log base 2 (x - 3), find the vertical asymptote by setting the inside part equal to 0. Then, find additional points by setting the inside equal to 1 and the base (2 in this case).* 01:13:22 🌐 *The domain of a logarithmic function is determined by the vertical asymptote and the highest x value. The range is always from negative infinity to infinity.* 01:14:33 🌐 *Exponential function domains are all real numbers, but their ranges are limited. For example, a horizontal asymptote at y = 1 limits the range to 1 to infinity.* 01:15:42 🔄 *For logarithmic functions like ln(x - 1) + 2, find the vertical asymptote by setting the inside equal to 0. Then, find additional points for graphing. The range is from negative infinity to infinity.* 01:18:51 📈 *When graphing log base 3 (2 - x) + 1, find the asymptotes by setting the inside of the logarithm equal to 0, 1, and the base (3 in this case). The graph reflects over the y-axis due to the negative sign in front of x.*

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thanks alot!

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I'm 7 minutes in and you are about to make me shout!!! Where have you been all my life!!!???

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Me too😂

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?!*

[@Matt-sc6gg]

Rihanna Where have you been all my life

[@zakinadhif]

I'm 7 minutes, and I'm still a baby.

[@placerinrome]

I spent 19 hours in 2 days trying to understand logarithms. YOU DID IT FOR ME! THANK YOU

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HOORAH DEVIL DOG!!!!

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Stanford Hurley

[@howmuchistoomuch]

I spent whole year

[@Freeaccs135]

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@@howmuchistoomuch are u still in school or college

[@pjapy]

This is the first time in my life I understand log. Touched my soul, I have no more math panic

[@workoutchannel4411]

yes broo when u understand math it really b touching I be in tears bc math used to make me feel defeated

[@jonathan-ww3sx]

Middle school teachers gave me math panic

[@madscientist1595]

Congrats you can now be a lumberjack. :)

[@spaceshipastro]

loll same

[@n0thmed116]

Can someone tell me why logarithms are so easy? Or is it just "The Organic Chemistry Tutor" that explains it soo well?... I bet.

[@nopda4095]

its becaues of the organic chemistry tutor fuck my doctor or professor or what ever a college teacher called mine cant even teach us the basics in the first week i just decided to ignore him and learn here and all my classmate did the same. even if i ask him a simple question he will make a fool out of me i asked him a simple question because he couldnt teach simple shit

[@ravennkings]

To be honest, if you know how exponents work, then it wouldn't take long to understand how logs work. the small number is the base number and the number beside it is the answer to the exponent. For instance, the log of base 6 and the other number is 216. as an exponent, 6^3=216. the only difference is the exponent is on the other side of the equal sign.

[@kimmarch90]

I failed my class the first time, I am currently retaking it (with the same horrible teacher that's available) and online tutors such as this guy, NancyPi and Brain Genius are teaching me what my "professor" fails to do.

[@beezalis9135]

@@ravennkings yooo thank you now it makes alot more sense🤯

[@tylera5598]

I think its that to understand logs you need to know how exponents and other algebra concepts work first, and by the time you get to logs its pretty much the same rules as any other equation but with an added rule. like converting logs to different formats makes sense because there's other problems where you do similar things

[@erikamferris]

You saved my life. My first Calc exam was a 31%, so I started watching these videos. My next exam I got an 87%. I'm taking the third one on Monday, and I feel like I'm gonna smash it. I can't explain how helpful these video are. Genius.... Pure genius...

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You prob did amazing, you will get that grade up from that 31%

[@tcggggg]

dawgg, Im in the same fkn boat I got a 38 on my first exam. Terrible you give me motivation bro cuz I need to hit all 90's to pass the class on my next three exams including the final..

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@@tcggggg You got this, I beleive in you

[@aaanawaleh]

I sat down for an hour and a half to roughly two hours studying this and I'm 37:00 in rn. Written 7 pages of notes. This channel is definitely one of the best channels in this genre on youtube. Great work and thanks for the content!

[@hildasnipes3486]

TIP: don't write notes for math, best way to get good is to spend hours listening to music and practicing pages of questions for every topic and practicing past papers to find what topics need development on.

[@Akihikoo_]

@@hildasnipes3486 you are right but sometimes you just don't find the past paper you want/need to practice that skill. But I agree with you, great point 👍

[@jakobkincade6856]

@@hildasnipes3486 I feel like its good to have a physical reference book of tips,tricks,rules, and functions so its can be easily referenced for the future or for a quick study or read up session.

[@hildasnipes3486]

@@jakobkincade6856 I agree, having notes with tricks and formulas is very useful, especially when you come to year 2 integration, however I find it much better to learn these tricks by applying them as much as possible in questions, i suppose it’s whatever works best for you, I achieved top grade without making any notes at all, which highlights that they are definitely not necessary

[@Tommy-gl3eq]

His "try this one'" would never let you get bored.

[@felicitypaigesings]

Legitimately i felt so discouraged and on the verge of tears because I couldn't figure out a way to understand and remember this topic. but this video is the perfect walkthrough at just the pace I need and covers alllll the tiny details that they fail to mention in class or other tutorials.

[@adesegunemmanuel8017]

You actually made me fall in love with maths

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same here

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i do math with love but i can t say i love it yet... i hope i acchive that one day!

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you are so whack.

[@JamxJam]

Even though it's been 6 years, I would like to just comment to send my appreciation for the content you provide for us to help us understand the lessons that we don't understand at all or need more understanding. Currently cramming for my exams tomorrow because I forgot to study, still haven't watched this video but I watched past videos and it helped me pass my exams last year hopefully going to pass this year.

[@WackyAhhPotato]

yep

[@tastetheengene8491]

I was literally crying panicking earlier because I have a test in two days on this chapter and I had no clue what we were supposed to do. 15 minutes in and I understand way more than I thought I would

[@dazzlingforests1373]

Haha that was me too. How did your test go?

[@quintenwilson1237]

I have my math final in an hour. I missed two weeks so I don’t know how to do any of this, meaning I have to learn this in an hour or I fail my math class.

[@tastetheengene8491]

@@dazzlingforests1373 I passed 👽

[@dazzlingforests1373]

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@@quintenwilson1237 how did it go?

[@kevinderooij2704]

man did what my math teacher was teaching us for the last week and a half and made no sense and this dude did it in 1 hour and a half, thank you for not making this math any harder than it should be teachers always try to make it more complicated and expect us to understand it fully immediately.

[@UPPERKEES]

Super well explained! Much better than Khan academy with the guy that constantly repeating the stuff while he writes... This is done very well!! Thanks!!

[@jsmnbre]

YES!!! so annoying lol

[@niroos6819]

but sal does explain like really well

[@osamaazam2382]

Yes but khan academy guy (Sal) gives intuitions and explains concepts. While he was more technique focused.

[@TheMrmuaz]

Still watch both, they save our lives lol

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Ahhhhhh how dare youuu

[@labibahaque9764]

I thought I had arithmophobia..coz I used to mess up easy equations every time. But now I realized I actually didn’t understand properly. If you understand math then it’s super interesting..I'm trying to get over my problems..this channel is like silver lining for me now.

[@backspaced]

I recently dropped to an F in "Algebra II/ Trig" class all because of the log unit we're having 😭 Thank you so much for this video because you just taught me what I was confused about in the weeks of learning this in class. I really hope my grade improves before the semester ends 😢 I really like your style of teaching Edit: It's been a while, but my grades did improve! 1st semester: B, 2nd semester: C (Not the greatest, but glad I didn't fail lmao)

[@zackxmagz6343]

How did you do?

[@backspaced]

@@zackxmagz6343 I'm improving a bit, my grade is now a C so at least I'm not failing math class haha

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that's me with pre cal rn lol

[@MrJ691]

This is me in Calc 3 rn that's why I'm watching these videos

[@backspaced]

@@shyrakrueger9693 Good luck 😭🤞 I barely survived last year

[@anthonydavidson6139]

I’m 33 minutes into this video and I’ve gone from a mumbling third grader to a rocket scientist. You are so good man!!!

[@nicolebordell8422]

I'm going back to college after 10 years and I had to take a math placement test. I hadn't touched math in years! Your videos saved me on my placement test! I was so worried I would have to take extra math classes, but your tutorials make so much sense and really helped me re-learn a lot that I've since forgotten. Thank you!

[@user-gg7bw9lg5j]

Good luck with your test!!!

[@nathanrobles6902]

The one day I am sick from school we go over logarithms that are on the final. Thanks for this video!

[@curtis2962]

damn lol

[@itsronelle6449]

same lmaoo

[@orangepinlac1949]

No most of the time his teaching method is just the same as in the school the difference is when you study at home with this video is You can sit in a good position or you can just watch it anywhere in your house, no distraction like classmates,the tutor in the vid is very friendly and the most important part is you can pause the video(tutor) and you can repeat it anytime and you can go back anytime you want 😂😂😂

[@curtis2962]

yeah and also in class the teacher can't go over one thing in depth for an hour (like he did in this video) they usually have a few chapters of stuff to go over in one class.

[@vagnerdpaula]

Agreed

[@kimmarch90]

I call bull. My professor does NOT go in depth at all. She expects you to know certain formulas and rules of math before taking her class and if you don't or can't remember then it "isn't her problem." Some professors just SUCK and do not help you when you ask for help. They find it tedious. I have only had a handful of professors that care to work with you.

[@walmart_dropout]

That moment this guy is better at teaching then my teacher thank you my dude!

[@Matt-sc6gg]

Almost a 90-minute video, but I'm absolutely hooked!!

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Sir, you.. you know what a hero looks like, what a teacher should be like ..Like you.... I wish i had a teacher like you when i was in 11th or 12th standard oh god give this man a very healthy and happy life ....

[@supp282]

I am studying to be an engineer and i find these videos much more insightful than my teachers lectures. sure my teacher is excellent at math but hes teaching it so casually like we already know everything about the subject...

[@borcf1557]

I can relate so much with you, I took trigonometry and the teacher was teaching us like we study the topic for multiple years and that we know everything he said

[@jimmybaez3695]

Please keep doing what you're doing, you're videos are amazing! Extremely helpful, Thanks!

[@emmanuelasuquo6482]

Nice explanation

[@AK20055]

Neither my book nor my teacher explained this in such detail. Thank you very much.

[@tonyomoto756]

You are awesome math teacher.. I learned new techniques in logarithm, natural log as well as trinomial factoring. I hope you can provide free handouts with more practice problems and answer keys in pdf from you, so I can practice more and learn other things like graphs of a functions for ln, log and exponents.. Thank your effort to impart your great math knowledge on us. More power!

[@sucaadxassan455]

But I didn’t understand the trinomial factoring😭

[@dugunu34]

This man is amazing. I learned more from him than I got from some of my teacher in one semester.

[@shyanmiller1449]

for years it took me to understand logs but when i found this video and worked them with you I really had to ask my self " is this really what i didnt understand?" thank you. you are an excellent teacher. keep up the good work

[@clementinedebeauvoir977]

00:01 evaluating log expressions 10:34 change of base formula

[@aishabah9401]

you are such an amazing tutor....i can't possibly stress how much you've helped me, especially for logarithms please continue what you're doing. It's sooooooo helpful

[@sadaftazwar2053]

Instead of my teacher teaching us about Log in 1 week. I just learned the whole topic about Log in 1 day from this video. Thank you very much!

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Please make a video about the systematic approach you take to teach. I would like to be a great teacher like you.

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Me clicks on this..... Studypubg ad appears Me: oh sorry I got this guys to teach me better for free

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God send angels in different ways. This tutor is surely one of them.

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this man is really saving my ap chem career and my precalc career at the same time

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Thank you so much for your videos, not only do you explain everything in detail, but you also provide a lot of examples for us to try solve on our own. For me this is the most important part, because I often understand the concepts in math but need to practice them alot to spot them faster. Thanks again keep up the good work!

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Who hates the study pug ads?

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Jus Tang ikr? Putting “pugs” in the name doesn’t make it likable, just barley bearable.

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Yup. These videos are teaching me just fine, I’m not going to use study pug. Don’t fix what isn’t broke!

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No lol it is good

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I do

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Litteraly the first 2 minutes ive understand more than what I've been trying to understand for months. Huge help thank you

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Ready for summer classes with my favorite professor, @The Organic Chemist Tutor. Best Covid Professor around

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I can’t thank you enough, you just saved me (and thousands of other students) on finals! Truly, thank you ❤

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Your explanation and detail is ABSOLUTELY brilliant. Your tutoring offers a huge sense of what resides in your heart. Thank u so much (Never too old to start learning again at age 56)

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Just searched up logarithms Organic Chem Tutor, and this most comprehensive video came up

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Thank you very much for creating such a fantastic comprehensive video like this. It truly helps in learning. The pace of each detailed explanation and how every concept flows into the other. Thanks again.

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Omg. I understood this immediately. Your teaching skill are amazing. 💕

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I'm always happy watching your videos Sir . You are helping a lot of students and I can't thank you enough

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Honestly, this channel has absolutely helping me throughout my GCSE's and until now doing my BSc and still come learn from here. So thank you Sir so much for your videos.

[@user-xh5we2wi5z]

I was suffering for a long time about logs , but after watching your lecture I'm now warm and dry. God bless you sir 🙏

[@julianaroh5509]

i did everything from start to finish.Thank you.

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Thank you for doing what my professor could not.

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i am from Sri Lanka, i leaned from you for my AL exam, now i passed the exam and now i come back to you for my university studies. thank you for doing this 🖤🖤

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I just want to say you helped me so much I started teaching myself math during covid with your videos and I just want to say you really make it easy to understand thank you

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1:16:22 THE LITTLE LAUGH THOUGGHHH

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JEEZ THAT IS WAY TOO CUTE!!

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You’re the reason I’m passing college algebra!!!!!

[@mercyjacob1]

This is the best tutor I have come across in my precalculus and calculus studies. Thank you so much. I even subscribed to monthly membership and it's worth every penny.

[@melimelon5368]

You are the best person to ever exist, not only did you go over all things logarithms but you also linked to tutorials for each section so thank you so much

[@ggpriya17]

You are awesome dude your the reason i have been getting A's in math

[@TheOrganicChemistryTutor]

Thanks

[@teresaajaw167]

this is the best online touter for logs..it really kinda helped me out..

[@schwiftyis350]

Thanks for teaching me what my Professor couldn't. I watched the whole video and did all examples twice.

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This guy helped me pass At Statistics and Probability, i know it may sound not too harsh but believe me i struggled.

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I recommend You this video kzbin.info/www/bejne/gH3boZSFo7yGj80

[@beacesar5739]

Thank you for this video! I never understood logarithm until now, you make math be really fun to learn. Keep going with this job, you're doing awesome! :)

[@nickhidalgo9818]

About to watch the whole thing wish me luck

[@available7829]

This has to be said, I love you for your teaching and have such immense gratitude as I was struggling with the content you have just made easy

[@seanwells1672]

Good stuff here.. came from pre and intermediate and currently in college algebra, thx to your videos!

[@sock9766]

Can’t explain how thankful I am for this

[@alexanderl947]

Thankyou for this video - Incredibly useful

[@ahtea725]

You're my new hero. I've decided to learn 'math' by pulling out the equations in Project Hail Mary and trying to understand them. First thing I had to learn was Logs. And BING! There you were. Thank you.

[@42luke93]

I disagree with the advertisement I got from some tutoring company. This video was completely useful and explained it way better than my teacher! Thank you.

[@happyfeet7123]

"If the base is larger than the number, it is a fraction" This is confusing since the log base 6 of 1/216 = -3 and 6 is obviously greater than 1/216. I think the correct term for this is the answer will be

[@weirdnet6925]

i got confused with this part of the video, what would the correct statement be then if you don’t mind me asking?

[@nuiyaroberts3495]

Because 6 to the power of 3 is 216, but the negative makes it a fraction which would be 1/216

[@nuiyaroberts3495]

And 6 is bigger than 1/216 ☺️

[@andrescientos]

@@weirdnet6925 he says that when the base is a whole number (rule: base>1) but the factor you want is a Fraction, in order to achieve that Fraction, the base's "n" will always be a number

[@agrimsingh001]

Getting a heart from the organic chemistry tutor is impossible

[@howmuchistoomuch]

Yes it is impossible

[@ItsJJOLO]

but learning logs in an hour isn’t

[@jan-niellouw1094]

After watching 100s of videos this is the best.... Good job. Love the video

[@lillatrollet87]

Wow, thank you for saving my azz! I struggled in class, but no more! Explaining everything in a simple manner without unnecessary complication of things.

[@x.v3n0m.x]

This helps a ton, and also he sounds like Cisco from Flash so it keeps me awake XD

[@kyle23011]

Second year engineering student, here to review the basics! Awesome video, great review.

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not me paying 10000+$ for tuition and textbooks just to come here and learn it better than ever for FREE! YOU ARE A LIFE SAVER

[@mysticblood212]

Thanks for all of this, I'm able to understand a lot of information by watching your video for 1 hour than spending several months in my maths class unable to understand most of the info my maths teacher taught me

[@mialanalene2182]

I’m literally three minutes in and I’ve learned so much about logs than I ever I learned in my math class at school 🤦🏽‍♀️

[@ritabernal2115]

The best prof in the whole world, specific and concise

[@user-ui7jc6qc8y]

You explained in details very well, thank you very much for this, i didn't quite fully understand this topic during my college days and now you made it CLEAR👏👏👏👏

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BEST TUTORIALS EVER!!!!

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I am in grade 8 understood the reality of math *Goes back to sleep in tension*

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Aadi sinha youll do this stuff in like year 11

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1 year again, then this'll be much ez for u.

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Come of senior year of high school it’s a whole different level of pain

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Wow thank you so very much. I spent a lot of time studying this video. Incredible work. Thank you for sharing your gifts.

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You taught me in an hour what my teacher would have taken 5 days! Thank you soooo muchh~

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Tell me why he teaches better than my paid college professor.

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I wonder what the math teachers in highschool has been doing that so many students come here to get supplementtary help. Maybe it's time for a reform in the education system.

[@afiazaman6622]

you are so descriptive with everything, Thank you soo much! My teacher is good at maths he is just not good at teaching weak students maths, he skips on soo many steps and sometimes its too fast and very confusing I wish he understood its hard for some of us, I wish he understood that it may seem easy for him and some of his best students but for the rest of us it gets tough

[@vishalthavare866]

i think one of the best teacher for maths i have ever seen thanks love from india

[@ThomasWSmith-wm5xn]

what software are you using??? I just want it for notes.

[@duck-bt6uy]

At 20:47 can the expression "1/3(2logx + log y - 4log z)" be further simplified? Because can't "log(x²y/z⁴)⅓ " be simplified as "log(x⅔y⅓/z⁴/³)" which is just "2/3log X + 1/3log y - 4/3log z"?

[@andrescientos]

There are several answers, but just read the questions to your assignments carefully to determine which answer to put. I also thought it was fully simplified at n = (1/3) but then he moves it to a =(1/3)

[@bondboy0563]

you are the lord and savior of knowledge. single handily carrying me through chem and functions for the past 3 years.

[@emilymorales611]

I wish this guy could be my instructor. You are amazing and your voice is not boring like my college professors. Easy to understand and follow along! I'm going to ace my test tomorrow!

[@erickmomanyi9439]

hello Sir, what is the difference for solving 'min.30:16 & min. 51:30?' kindly help. thank you.

[@nevilpepaj9772]

No

[@Awesome-ct7vr]

In the first 1 he is not solving it but showing the difference and transformation between logarithm and exponential form, how they become one another. In the 2nd reference, we see a log that in its exponential form equals a number to the power of a number that isnt an integer. Therefor you cant use simple knowledge and logic to solve for x and have to get a precise value. Hence convert the equation to natural log and isolate the x. Once thats done plug the natural logs to the calculator and get what x worth aka as the missing exponent

[@ethanhuynh8772]

10:26 he says the answer is a fraction when the base is greater than the number in the brackets, but why isn’t log6(1/216) a fraction?

[@mielatyonas6254]

log6(1/126) Log6(1/6^3) Log6(6^-3) Answer is -3 not fraction

[@mahnoorshahid4508]

It was an amazing video. Learned so much more in an hour an a half than i did in 3 lectures my professor gave. God bless you!!!

[@OneGoodWoman84]

Baby!!!! You just don't know how much I appreciate you! The way you broke that down may have saved me in this Calculus class........ oooh your heaven sent. Must Be! Thank ya! Won't he do it?