The INTUITIVE way to SOLVE ANY Exponent Equation! | 4 Minute Math Tutorial!

[@MathMasterywithAmitesh]

Check out kzbin.info/www/bejne/p5OkaGSDg9B-q80si=nzH7Yq7NLk6aWzu9 to see a simpler exponent equation solved in 3 mins, where logs are avoided altogether! 😊 Subscribe for math mastery across all levels and topics! 🥳🎉🎊

[@PiyushSharma-u6v]

Can't this question be directly solved by taking the log on both sides in the beginning? By taking log on both sides in the beginning, value of x will be equal to (log 4 base 6 + 1)

[@MathMasterywithAmitesh]

Hi @PiyushSharma-u6v thanks so much for your comment! Yes, you can write the final answer in a few different ways, for example, if you take log to the base 6, then you would obtain the answer you mentioned, and it is perfectly correct. 🥳 I already addressed the reasoning for my approach in a reply to another comment on this video, so please check that out (clicking on the following link will highlight my reply): kzbin.info/www/bejne/oqmtZ32rfqd0iZY&lc=UgxVB1NgWgXhtg26To54AaABAg.ACOublYnWvSACPqjI6hGdT I hope you have an amazing day/evening/night! 😊 I wish you a safe and wonderful holiday season too! 🥳 🎉 🎊

[@EC4U2C_Studioz]

I would have just solved this equation with a single step and left it in log form. Taking the log of the matching base automatically cancels out the matching exponential base, leaving only whatever is in the exponent. In this case, it would be log of 24 to base 6.

[@MathMasterywithAmitesh]

Hi @EC4U2C_Studioz thank you so much for your comment and for sharing your approach! I am always happy to see your comments! 😊 Your answer is correct, of course, and we can just say at the beginning that x = log_6(24) since the definition of log_6(24) is: "the value x such that 6^x = 24". The point of the video is to give a simplified version of this answer in terms of log_2(3). The reasons for this are: (1) of course, to practice with exponent laws etc but more significantly (2) to show how lots of logs can actually be written in terms of a small number of simpler ones. For example, if you had an equation like 18^x = 24 or 6^x = 48 or 2^x = 6 or 3^x = 16 etc., you can use the approach in the video to write all the solutions in terms of log_2(3) only! (The reason is that these numbers, 18, 24, 6, 48, 2, 3, 16 etc. only have 2 and 3 as prime factors.) In particular, if you know just this one value log_2(3), you can solve lots of equations immediately! 🥳 From the point of view of math before computers/calculators, people used to have log tables where they had a few simple values noted down (like log_2(3), log_2(5), log_3(5) etc.) to several decimals, and could then solve lots and lots of equations using just these small numbers of values. If we just write the answer as log_6(24), we would have to basically learn a different value for each equation. I hope you have an amazing day/evening/night! 😊 Happy Holidays to you!!! 🥳🎉🎊

[@EC4U2C_Studioz]

If leaving the answer in log form is accepted, the log of 24 to base 6 would have been ok. I would prefer to reduce logs if the log evaluation is an integer. For example, I would not have left log of 8 to base 2 as it is as it evaluates to an integer response of 3.

[@MathMasterywithAmitesh]

@@EC4U2C_Studioz Hi! Thank you so much for your comment! 😊 Yes, I understand your point, that log_6(24) is a perfectly correct answer, and with a calculator/computer, we don't need to simplify to know what this is, so we can practically leave it like this! However, I think it is still fun and interesting to do so, and we can learn more about logs by doing so as well! 🥳

[@WolfgangFeist]

Why not take a 'general' log at the beginning, factor then and decide at the end which basis is most convenient? (:-) My feeling: less complicated, you don't have to 'collect' 2 times (fist time by basis, secend time for x - which was at the left site already at the beginning).

[@MathMasterywithAmitesh]

Hi @WolfgangFeist thank you so much for your comment, as always, and for sharing your thoughts! 😊 Yes, I agree with you that your approach is simpler than the one in the video. The (entirely stylistic) reasons I didn't use this approach in the video are because: (1) I wanted to show how to manipulate exponents and exponent laws to see what is going on behind the scenes of the log (i.e., avoid log laws as much as possible). I think one more thing I could have done in the video in this philosophy is to take fractional powers once we had 2^{x-3} = 3^{1-x} to write 2^{(x-3)/(1-x)} = 3, and then it's clear (x - 3)/(1 - x) = log_2(3) by definition and we don't need any log laws. (2) I wanted to get the final answer in terms of log_2(3), and to show why this is natural in this problem/see why that arises (which may be more intuitive than natural log, for example). I wanted to emphasize the idea of writing 6^x and 24 with the same base as a natural first step. However, this is not to say I think the video's approach is better, I actually think yours is much more efficient for solving the problem for the reasons you gave. I think I should do a video assuming good knowledge of log laws with the approach you mentioned. 😊 I hope you have an amazing day/evening/night! 😊 Happy Holidays to you and your family!!! 🥳🎉🎊

[@WolfgangFeist]

@@MathMasterywithAmitesh Thanks - I understand. In these cases, there we have e.g. 3 terms (a^x+b^x = c^x), the "log first" strategy does not work (?). On the other hand, these problems only work if there is a simple solution to the resulting polynomial equation with u=d^x, so these seem to be backwards constructed problems. Is there an approach to "guessing" a first solution for polynomial equations with higher exponent (>2)? (My preference might be due to being trained in logarithms early on (in the 60s, calculators were not yet widely available). I see in my courses that younger generations are not familiar with logarithm rules - a problem since some areas of physics and engineering still use widely used logarithm scales, like dB or pH -values.)

[@WolfgangFeist]

@@MathMasterywithAmitesh One more question: next year we have to create a building physics course for non-academics. I realize that I have to introduce some mathematics - like "numbers" (well, decimals might work), variables, simple concepts like (a+b)², simple equations, roots, ... and also elementary integration and derivatives. Is there a source I could recommend? Many of the participants may think "we don't need theory - we are practical people"; I try to convince them in the sense of "nothing is more practical than a good theory" (there are some good examples in physics, like the kinetic theory of heat).