solving a logarithmic equation with different bases

How do we solve a log equation with different bases? Here we will see how we can use the change of base formula for logarithm to solve log_4(x)+log_2(x)=6.
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[@raphberry]

Could you just have done: ln(x) + 2 ln(x) = 12 ln(2) 3 ln(x) = 12 ln(2) ln(x) = 4 ln(2) = ln(2^4) = ln(16) x = 16

[@vitakyo982]

Too simple is not chinese ...

[@bitterberry_]

Even don't need ln log2 x + log4 x = 3log4 x = 6 log4 x = 2 x = 16

[@harshvirgrewal2403]

That’s what I was thinking, just add the ln(x)’s and then divide both sides and it’s simple from there

[@BigDBrian]

@@bitterberry_ This particular example works out nicely like that due to the bases being 2 and 4. But I think the point is to show a more general method

[@rosebuster]

@@BigDBrian You should always use the simplest approach for the task at hand!

[@perlnut]

Why not recognize 4 is 2^2 and change log4(x) to log2(x)/log2(4) and continue from there? Much simpler.

[@rosebuster]

Yeah, I also think it makes more sense to be a little bit more picky choosing the log base when you can choose any base you want. Better think what is the simplest one to use instead of sticking to natural logarithm just because it's the one you're most used to. I always say it's better to think a little instead of following formulas mechanically. And personally I favour base 2 logarithms anyway, because they're the ones you'll find most commonly in computer science. :)

[@morksan9973]

Mathmaticians never seek for easy way

[@tbg-brawlstars]

Or just use that property when power of base goes in denominator 🙂

[@nemanjalazarevic9249]

Theres also the short cut formula of: Log x^n(y)=Logx(y)/n

[@tbg-brawlstars]

@@nemanjalazarevic9249 yes, it is a basic log property

[@donati880]

Why not : 1/2log2X+log2X=6 3log2X=12 log2X=4 X=2^4 X=16

[@NasirKhan-lq5jl]

Exactly what I did in my mind when I saw the thumbnail.

[@ivansavchuk7956]

+)

[@NTENewTechnologicalEducation]

donati880, that is also correct.

[@Viesto1980]

Exactly,

[@aman-kr]

Yes this is very easy and simple method.

[@animalfarm7467]

Just a thought - If you use a base change to "2" instead of "e", you rapidly get to (log2(x))/(log2(4)) + log2(x)=6; 3/2*log2(x)=6 or log2(x)=4, hence x=2^4

[@johngreen3543]

You have the "best" way to do it

[@xtree2817]

Love ya

[@cipherbenchmarks]

Where u get 3/2

[@animalfarm7467]

@@cipherbenchmarks : Factor out the log2(x)) from (log2(x))/(log2(4)) + log2(x) (log2(x))(1/(log2(4)) + 1) and as log2(4)=2, (log2(x))(1/2 + 1)=(3/2)log2(x) Hope that helps.

[@manka5464]

@@animalfarm7467 I appreciate you responded after three years and yes, it helped

[@kujmous]

I like this video especially because I can see the answer beforehand. I did it differently, but it uses the same reasoning. LOG4(x) + LOG2(x) = 6 LOG4(x) + 2(LOG4(x)) = 6 3LOG4(x) = 6 LOG4(x) = 2 x = 4² = 16

[@yashjakhmola]

Not bragging, but this was too easy. I literally solved it in my head just looking at the thumbnail. Your videos are awesome tho.

[@SHASHANKRUSTAGII]

Same

[@ridditdit2686]

It's still really cool math so I don't mind

[@shacharh5470]

Same

[@sangeetanarendrasingh5416]

Yeah, same

[@SartajKhan-jg3nz]

Easy for us in higher grades. Consider the people who are not familiar to the concepts of logarithms🙃

[@lostwizard]

As others have noted, observing that the bases are both powers of 2 suggests that the best base to change to would be 2. That then means the denominators in the change of base become nice integers and the result then trivially simplifies to log2(x) = 4. Of course, your way works more generically and shows off more log properties.

[@orisphera]

I think base 4 would be even better here

[@AndreasChristianto]

i prefer: ln x + 2 ln x = 12 ln 2 3 ln x = 12 ln 2 ln x = 4 ln 2 ln (x) = ln (2^4) x = 2^4 x = 16

[@hollow6079]

same here, just think it's unnecessary to bring the 2 into ln x

[@flamingpaper7751]

You can do that last step a little easier. When you have ln(x) + 2ln(x) = 12ln(2) You can simply add the ln(x) on the left since ln(x) acts like a variable and get 3ln(x) = 12ln(2) Then divide both sides by 3 to get ln(x) = 4ln(2) Then put the 4 into the ln(2) to get ln(x) = ln(2^4) Then x = 2^4, which is obviously 16

[@not_vinkami]

Maybe it's time for us to make it *COMPLEX*

[@mathsplus01]

I really enjoyed how you showed how easy to change the base. Great!

[@HeyKevinYT]

ln x and 2 ln x are like terms

[@blackpenredpen]

Kevin vs. Gamingz yup! It was 10:30pm when I recorded this..

[@tbg-brawlstars]

BlackPenRedPen : Takes 6 mins to solve the world's easiest log question Me : Solves it in literally 10 seconds in head

[@blackpenredpen]

LITERALLY?

[@tbg-brawlstars]

@@blackpenredpen According to google Literally definition : used for emphasizing something

[@mulezichanje]

He's tutoring, you're not.

[@kirbo722]

One of my favorite bprp videos♥️

[@ib9rt]

At 3:18 I was so stunned about what you did next that I had to check the calendar to make sure it wan't April 1st...

[@Shreyas_Jaiswal]

Haha, it could be written just 3ln(x).

[@boldandbrash2499]

I have just applied 2 to base of logarithm and saw that (log2^x/2) + (log2^x/1) = 6 And then i have multipled second term by 2 then saw that (log2^x/2) + (2log2^x/2) = 6 this turned into 3log2^x/2 = 6 and did the equation which turned into log2^x = 4 and simply found the x equals to 16. I have watched so much of your videos. And w/o you i couldn't be able to find out the answer. All thanks to you

[@manuelgonzales2570]

Very good exercise. Thank you!

[@abu-karz]

So perfect so beautiful i can't believe it. Never seen something so cool in maths with such hard functions like this.

[@denismilic1878]

I rarely can solve math problems on this channel in my head, or even solve without pain but this problem is super simple or I got some wierd inspiration. Answer just poped in my head Sheldon style. log4(x)+log2(x)=6 log2(sqrt(x))+log2(x)=2 +4 log2(sqrt(x)) = 2 and log2(x)=4 x= 2^4 = 16

[@makhloufbenmehiris9559]

You are the best professor in life Thank you

[@thomasg6830]

ln(x) + 2×ln(x) = 3×ln(x)

[@terapode]

You are a very good teacher.

[@kidtherookie6019]

I found another way! Note: 2*log_4(x)=log_2(x) log_2(x)+2*log_2(x)=12 log_2(x^3)=12 x=2^(12/3)=16

[@angelmendez-rivera351]

Here is a good one: can one use the Lambert W function along with all the other elementary functions we are acquainted with to solve an equation of the general form e^x = Ax^2 + Bx + C ? And if so, what is the general solution in terms of A, B, and C?

[@Thefarnworth1]

You are amazing! Thank you so much.

[@suniltshegaonkar7809]

Also that we can convert Log X_4 = 1/2 of Log X_2. So both terms are at Base 2, taking common the Log portion). 3/2* LogX_2 = 6 >>> X^3/2 = 2^6 >>> X= 16.

[@fr0si834]

I did the same thing , is it a right way to solve logarithmic equations that can be converting to be the same bases ??

[@ja1212az]

Amazing work man

[@PedroHenrique-vs3mf]

Incredible work

[@BigDBrian]

when you got lnX + 2lnX = 12ln2 you should just add the lefthand side, then divide everything by 3. So you'll get to lnX = 4ln2 = ln16 a lot faster and don't have to take the cube root which creates extraneous solutions

[@oldravianspublicschool7437]

Good example for understanding the logarithms' law of change of base

[@hailmary7283]

Instead of doing ln(x)/ ln(4), why not just do log2(x)/log2(4)? Then you get log2(x)/2 + log2(x) = 6. Then we have (3/2)log2(x) = 6 log2(x) = 4 x = 16 This just seems a lot easier. Love the videos.

[@nvapisces7011]

This one was easy. Could also use substitution to simplify things even further after change of base

[@bobus_mogus]

Awesome video!

[@NeedBetterLoginName]

Around 3:30, you could just do ln(x) + 2ln(x) = 3 ln(x) and go from there, quicker and simplier

[@alihasani648]

These videos are the only revision I do

[@josephdeyoung8754]

It’s crazy how many math problems I do and I remember doing hundreds of these just months ago and I never retain the rules

[@chyawanprash]

I did it like this. Let y=log4 (x) Let z=log2 (x) y+z=6 4^y=x 2^2y=x 2^z=x 2^2y=2^z 2y=z y+2y=6 3y=6 y=2 log4 x = 2 x=4² x=16

[@nepasdisponible]

Nice.

[@sergioh5515]

I see that spiderman is a patron XD

[@yasharora3797]

You can use the property where log x to the base b^n is equal to 1/n times log x to the base b.

[@tombombadil1351]

name of that property?

[@Tomaplen]

What is log_i(i!¡) ?? (Logarithm with base i of i-th tetration of i subfactorial

[@michaelschneider4385]

Love it buddy

[@onelastmanstanding]

YOU ARE STILL SAVING LIFES

[@GirishManjunathMusic]

Given: lq(x) + lb(x) = 6 Knowing lq(x) = ½lb(x); as lb(4) = 2: ½lb(x) + lb(x) = 6 Adding fractions: (3/2)·lb(x) = 6 Multiplying both sides by ⅔: lb(x) = 4 Exponentiating both sides using 2 as the base: x = 2⁴ x = 16. Here, lq(x) = log(x) with base 4, lb(x) = log(x) with base 2.

[@lorenzoluccioli6699]

Could you demonstrate the property you just used? (The one to make any log the base that you want)

[@dm_saj1119]

Nice job sir

[@mikenotpike]

At 3:30 just simplify to 3ln(x)=12ln(2) and divide both sides by 3 to get ln(x)=4ln(2), simplify to get ln(x)=ln(16), cancel the 'ln's and you get x=16

[@mjones207]

I think this is easier to use the change of base formula on just the first addend only using log(base 2). This gives log(base 2)x / 2 + log(base 2)x = 6 → (3/2) log(base 2)x = 6 → log(2)x = 4 → x = 2⁴ = 16.

[@abhirampaku6070]

Easier method, log_4⁡x + log_2⁡〖x 〗=6 log_(bxb…..ntimes)⁡〖a 〗=1/n( log_b⁡a ) 〖1/2 log〗_2⁡x + log_2⁡x =6 3/2 log_2⁡x =6 Dividing and multiplying both sides with 3 and 2 respectively log_2⁡〖x 〗= 4 log_b⁡a=x then a=b^x Therefore, x=2^4 =16

[@rafikhan908]

Another option: 3lnx = 12ln2 ; divide by 3; rewrite RHS as ln2^4 and then antilog...etc.

[@adrianfrauca8118]

3:25 alternatively, lnx + 2lnx = 12ln2 => 3lnx = 12ln2 => lnx = 4ln2 => x = e^(ln2*4) => x = 2^4 => x = 16

[@ziasquared5753]

Not really a math question, but how do you write on a whiteboard using two pens? I kind of wanna see a close up lol.

[@ffggddss]

Easy after a little tutelage in legerdemain. Manipulating multiple objects in one hand. Fred

[@londonalicante]

Everybody else saying there's a simpler way. My issue was that there was an unnecessary JAZZ PIANO playing.

[@hata6290]

Oh my god I figured out a bprp problem before he finished on a three year old video about algebra 2 LETS GOOOO

[@di-riso]

You could also have taken log 2 getting log 2x +2logx=12 then log 2 x • x^2 so then log2 x^3 and then proceed the same way

[@ericventalon6113]

in 4th line we can also write: ln x + 2 ln x = ln x (1 + 2) = 3 ln x = ln x ^ 3 I love this vidéo it s still good.

[@MrConverse]

Why not add ln(x)and 2ln(x) to get 3ln(x) then divide both sides by 3? Much simpler than the route to the solution shown in the video (IMO).

[@ChaineYTXF]

I agree, but it has the advantage of showing more rules on logarithms

[@runeboas6421]

Remember he teaches stuff.. He wanna use as many different methods he can, without confusing people :>

[@MrConverse]

I agree, Delta & Rune, that he probably did it the way he did to include more log laws. But I think taking the simpler route here would have been better. Those other rules could have been shown in another problem (or video). ;-)

[@blackpenredpen]

Thad Spreg Honestly... I totally forgot about it. It was 10:30pm when I recorded this at school after 5 hours of teaching...

[@MrConverse]

blackpenredpen you work very hard. We appreciate all your efforts!

[@roger7341]

To avoid confusion I would just solve log(x)(1/log4 + 1/log2) = 6 for log(x) and then set x = 10^log(x) = 16, where log is to base 10.

[@jinishtrivedi2429]

Sir you could directly write.. Log[4]x=(1/2)log[2] HERE I USED SQUARE BRACKETS [. ] TO DENOTE THE BASE OF THAT LOG This could become more easy

[@AbdulMoiz-uj3wg]

Can you make a video on modular arithmetic?

[@cyberguardreal]

I'm quite new to logs but realised that log_x(y) is the same as log_(x^n)(y^n) so you can rewrite log_4(x) as log_2(x^1/2) and proceed to solve log_2(x * sqrt/x) =6

[@roderickwhitehead]

I woulda just combined the terms on the left to 3*ln(x), then divided both sides by 3, yielding: ln(x)= 4*ln(2)

[@MartinUToob]

I really liked the conclusion. (That must make you feel real smart.)

[@elektriksvarsiki]

Log2(x)=2log4(x) so, 3log4(x)=6 Log4(x)=2 By definition,x=16

[@keshavkumar-ze6gr]

Sir.can u plz make a video on wallis and gamma in integration ....it would be of great help ....plz

[@masterclash9959]

A “hidden” rule I found with logs is that if you have log_b(a), it will be equal to log_(b^2)(a^2) or alternatively 2log_(b^2)(a). This is because if you use your change in base formula and exponentiate the arguments to the same number, then you can use the power rule to take the exponents out of the argument and cancel, of which would leave you with another fraction to apply the change in base formula. Example: log_9(4) ln(4) --- ln(9) ln(2^2) --- ln(3^2) 2ln(2) --- 2ln(3) Simplify to ln(2) --- ln(3) And log_3(2) Hence, log_9(4) = log_3(2)

[@rw5035]

Excelent!!

[@PracticeMakePerfectMuslim93]

this shown step by step rather than shortcut 😊

[@ffggddss]

log₄x + log₂x = 6 Well, because 4 = 2², we can write log₂x = 2·log₄x And then we have log₄x + log₂x = 3·log₄x = 6 log₄x = 2 x = 4² = 16 Note that the exponent, 2, in 4 = 2² is the log base 2 of 4: log₂4 = 2. So in more general cases, when there are logs of more than one base, they can all be converted to a common base, using logᵤx = logᵤv · logᵥx You can choose any base you want as the common base - whatever works best. Now to watch Mr. Pen, to see how he does this . . . Ooh, yes that's the right answer, but sorry to have to say I don't think this is the best way to get there. Once you have: lnx + 2·lnx = 12·ln2 just combine the LHS to get: 3·lnx = 12·ln2 lnx = 4·ln2 x = 2⁴ = 16 Kudos for giving the change-of-base formula - that's a powerful tool in log problems! Fred

[@tobiaschapinda6771]

Logarithmically Transform the function y=(〖(3x^2+2)〗^2 √(6x+2))/(x^3+1)

[@user-vk4hv8cf6k]

Thanks

[@MostafaAhmed-kn4yy]

another solution : Suppose that log4(x) = y If 4 ^ y = x - - or 2 ^ 2y = x We take log2 (x) in the last relationship If 2y = log2 (x) then we substitute the original equation to find y y = 2 After compensation, we find the value x x = 16

[@viktyusk]

You chose the most complicated way to solve this.

[@YamaKangaroo]

I know we like to use the natural log for the general case, but for this specific case, wouldn't it have been simpler to convert the log_4 to log_2?Then you would have had (1/2)log_2(x) + log_2(x)

[@fr0si834]

I use it and I get the solution X=8*(cubicroot of 2) "its soo close to 16" and I get the same solution when I use the Mathway website . Am I get the right solution ??

[@mcwulf25]

Easier to keep the coefficients outside of the ln. 3lnx = 12ln2 then x=2^4.

[@dkiproch]

At second step if you just set the equation as log2(x)/log2(4) + log2(x)/log2(2) = 6 It would be much easier since log2(4)=2 and log2(2)=1

[@lionbryce10101]

Wow, I'm seeing a lot of different ways to solve this. Let me throw my 2 cents in (btw, if you do it this way, you'll lose points on most exam) See both are powers of 2 Try 8 as an input, see that it's short Try 16, it worked.

[@shreyanssharma5280]

Just have log2(x) as "t" and then t+t/2 = 6 which gives t= 4 so log2(x) = 4 or X= 16.

[@tobiaschapinda6771]

Given that log_a ⁡p=0.7and log_a ⁡q=2, find log_a⁡〖p^2 〗, log_a⁡〖p^2 〗 q and log_a⁡ (apq)

[@deyomash]

I recognized that since we have logs of 4 and 2 which can both be written as a number base 2 I let x = 2^m. Then you get m/2 + m = 6 or m = 4 so x=16 :P

[@tooba6290]

We could also resolve the first term into base 2 it's easier that way

[@liamtanner337]

Isnt there also 16e^i*2pi/3 and its conjugate if we talk complex?

[@mosesmwata7613]

Correct but there’s still a simpler way to solve it log2 (x^0.5) + log2 (x) =6 log2 (x^1.5) = 6, we know log2 (2) =1 Let introduce log2 in the second member log2 (x^3/2) = 6log2 (2), (we know 3/2 = 1.5) Let’s write 6log2 (2) in the form of exponential log2 (x^3/2) = log2 (2^6) Then the log2 canceled out X^1.5 = 2^6 X= 2^6/1.5 X= 2^4 X= 16

[@YosefTesfay]

I think this method is also possible! Log 4 (x) + Log 2 (x)= 6 (Log 2 (x))/ Log 2 (4) + Log 2 (x) = 6 Let t = Log 2 (x) t/ Log 2 (4) + t = 6 t/ 2 + t = 6 3t/2 = 6 t= 4 Log 2 (x)= 4 x = 2^4 x= 16

[@MrRyanroberson1]

here's how i solve it: 4_x means log base 4 of x, so we have the identity 4_x = 1/x_4 = 1/2 x_2 = 2_x /2 so 4_x + 2_x = 3/2 of 2_x = 6, we get 2_x = 4, x=16

[@beggibob6261]

It took me 5 steps to finish this and i feel like he makes it complicated, i mean why don't you do like this for quicker? Step 1: we can turn the question into 1/2log2(x) + log2(x)=6 Step2: log2(x)(1/2+1)=6 Step3: 3/2 log2(x)=6 Step4: log2(x)=4 Step5: x=16. DONE!

[@Iamaayushmaurya]

Please integrate tan inverse X whole upon 1 + x square to the whole power 3 by 2

[@anglo2255]

I always wanted logs for my birthdays”. Also in high school algebra I swear they taught us to”anti-log” both sides when you wanted to “e” both sides. I know we didn’t do that in this video. Just made me think of that.

[@bearme1160]

Just notice log4 is always a half of log2 and it can be rewritten as log4+2log4=6 which then can be written as 3log4=6-> log4x=2 x=4 to the power of 2=16

[@wahyuadi35]

This logarithmic question is still so.. easy as cake.

[@abdulrahmanelmoussa5921]

Please can you simplify Arccot(cosx) Inverse cotangent of cosine X

[@generaldarian1263]

Another way I found to solve this: From lnx/2ln2 + lnx/ln2 = 6 Factor out (lnx/ln2): (lnx/ln2)(1/2 + 1) = 6 (lnx/ln2)(1.5) = 6 lnx/ln2 = 4 lnx = 4ln2 x = 2^4 = 16

[@TheSandkastenverbot]

I just went by the definition of the logarithm: 4^2 =16 = 2^4. So if x=16 then log_4(x)=2 and log_2(x)=4. The sum is 6. This, of course, worked only because x is an integer and the logarithms as well.

[@StaticBlaster]

Easy peasy!

[@inankazanc2724]

So this equation contains 3 solutions,since x^3=2^12?

[@angelmendez-rivera351]

No, this equation is not a cubic equation, where are you getting the idea that this equation is a cubic equation? (3/2)log2(x) = 6 => log2(x) = 4 => x = 2^4 = 16. Turning it into a cubic is not only unnecessary, but it changes the equation itself, and therefore it’s solution set. You are not allowed to simply create irreversible functions in a manipulation.

[@overlordprincekhan]

You could take log base 2 instead of natural log. That would simplify more

[@ricardovillanueva6519]

Dear Jenna Grace, log[(4x)(2x)]=6. Now this: log(8x^2) = 6, a base 10 logarithm. Change it to exponential 8x^2=10^6 ; x=sqrt of 10^6/8

[@holyshit922]

I would choose 4 or 2 for the base because in these cases i can use change of base formula only once

[@jeffreyluciana8711]

Wonderful

[@i_am_anxious0247]

ln(x)+2ln(x)=12ln(2) 3ln(x)=12ln(2) ln(x)=4ln(2) ln(x)=ln(2^4) x=2^4 x=16 It’s mathematically the same, it just feels better doing it this way for me idk

[@vhmix379]

log(x)/2log2 + logx/log2 = 3logx/2log2 = 6, 3logx =12log2, logx/4log2=1 , logbase16(X)=1 so that means x =16