Harvard University Pure Mathematics Course Entrance Exam ! ✍️🖋️📘💙

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@clivetooth 8 күн бұрын

A simple approach is as follows. We can see a little more clearly what is going in if we solve the more general equation a^b^m = b^a^m. The logarithm base can be any convenient number: e or 10 or a or b etc. a^b^m = b^a^m b^m log(a) = a^m log(b) [taking the log of both sides] b^m/a^m = log(b)/log(a) (b/a)^m = log(b)/log(a) m log(b/a) = log(log(b)/log(a)) [taking the log of both sides again] m = log(log(b)/log(a)) / log(b/a), and we are done. = (log(log(b)) - log(log(a))) / (log(b) - log(a)), if we prefer. = log(log(b)) / (log(b) - 1), if we take logs to base a. The fact that, in the given question, a and b are both squares is fairly irrelevant.

@DuettoSpider 8 күн бұрын

Yes the procedure is correct, m is approximately equal to 0.57. Be careful not to get confused as I did when checking the result. In fact 9^4^m in this case should be written as 9^(4^m). So when checking, first carry out 4^m, and then raise the 9 to the obtained result. In fact it is not a power of a power (which instead would give 9^(4m) which however gives another result)

@superacademy247 8 күн бұрын

I'm glad you found away to understand the process of arriving at the correct answer 💕🤩🤩✅💯💪

@DuettoSpider 8 күн бұрын

@@superacademy247I've always liked math, I really like these tests. I always want to understand things and look for the right answer. If something doesn't add up, I keep trying until I understand.

@martinphipps2 4 күн бұрын

4^mlog9=9^mlog4 so (4/9)^m=log4/log9 so mlog(4/9)=log(log4/log9)

@superacademy247 4 күн бұрын

Your approach is incorrect ❌. Follow the method in video ✅

@alberta1783 8 күн бұрын

This is a very basic operation.

@fortechnicalstudents0.866 4 күн бұрын

M=0.66666=0.7

@isaactate1667 5 күн бұрын

M=0

@SYAN23SYAN13 4 күн бұрын

Tanks for watching

@superacademy247 4 күн бұрын

Glad you enjoyed it! 🔥🥰✅💕

@ciaucia156 8 күн бұрын

=LOG(LOG(3,2),3/2)/2 or =LOG(LOG(2,3),2/3)/2 =LOG(LOG(3;2),9/4) or =LOG(LOG(2,3),4/9) and the "simpliest" =LOG(LOG(9,4),9/4) or =LOG(LOG(4,9),4/9)

@soudhrizpal1056 3 күн бұрын

Put m= 1/2 in question......LHS≠RHS

@raimondobalestra9085 9 күн бұрын

Hello! It could play the unknown "m", because I get different results. Thank you.

@DuettoSpider 8 күн бұрын

in fact the value of m found is not correct. The procedure seems correct, but in reality it does not work. The solution is m = 0

@clivetooth 8 күн бұрын

@@DuettoSpider No. m=0 immediately leads to "9=4", which is false. You may have mentally bracketed the question wrongly. A^B^C always means A^(B^C), not (A^B)^C.

@DuettoSpider 8 күн бұрын

@@clivetoothhi...yes you're right...I was wrong when I checked the result. I also solved it by applying the logarithm twice, obtaining m = 0.57 approximately. But then when I checked the result I mistakenly considered 9^4^m, a power of a power. Instead it is only 4 raised to the power of m, and not also the base 9 (in fact I finally had a doubt and tried the calculation with m = 2 and actually I noticed the error). That's why in the end the check didn't add up. Anyway thanks for your answer. Yes of course m = 0 is not a solution because I would get 9 = 4 which is not true. I tried to re-insert the correct form on wolfram alpha, but it actually can't calculate it due to exhaustion of the computing resources (maybe it could calculate it with the paid pro version)

@clivetooth 8 күн бұрын

@@DuettoSpider Hi. I am glad you understand it now. Btw, I typed 4^9^0.567941283958 - 9^4^0.567941283958 into Alpha and got -7.42685 × 10^-11. I am not sure why you had a problem with Alpha.

@DuettoSpider 8 күн бұрын

@@clivetoothhello sorry, there must be an error in your setting. If you enter 9^4^0.57 on alpha it must give you 126.756. You must get the same value on the second member. So the difference you indicated between the first and second member must give you 0, since the first member must be equal to the second member; not the value you indicated. The value of m that was found with the double logarithm procedure, guarantees this equality (that is, it is said that the solution found verifies the given equation). On wolfram alpha try to set the equation using parentheses in this exact way, so as to avoid ambiguity: 9^(4^m) = 4^(9^m)........you will see that wolfram alpha will not be able to perform the calculation due to excessive request of computing resources (maybe you can perform the calculation if you have access to the paid pro version and not free). Keep in mind that wolfram alpha does not solve the equation with the classic steps that we do, but solves it at a numerical level. Wolfram alpha in the free version cannot finish the calculation because it has a limited calculation time available. The pro version has three times the time available compared to the free version.

@SidneiMV 10 күн бұрын

4ᵐln9 = 9ᵐln4 2²ᵐln3 = 3²ᵐln2 (3/2)²ᵐ = log₂3 2mln(3/2) = ln(log₂3)

@MulaSlayer 9 күн бұрын

With all due respect my brother, 9^4m will always be odd regardless of m and by the same logic, 4^9m will always be even, a bit like equating apple and orange. I don't think you should take Oxford and Cambridge very seriously when it comes to mathematics, philosophy and cultural history.

@superacademy247 9 күн бұрын

Your definition is based on integers. But as for this question no integer conditions are set at the beginning!

@shahzaid1765 9 күн бұрын

WOW such confidence it takes to call one of the best universities in the world impotent. fucking clown

@jv_16 7 күн бұрын

Stop trying to be a smartass and use your brain

@MarioRBSouza 9 күн бұрын

Muitas passagens desnecessárias. Harvard não necessita de tantas explicações.

@superacademy247 8 күн бұрын

I'm glad you found the solution helpful! 💯🙏🤩💕Thanks for your feedback! 💯🙏🤩💕

@Vega1447 8 күн бұрын

m = 0.567941283958152 Just take logs twice... And no, not a Harvard exam.

@Vega1447 8 күн бұрын

@DuettoSpider au contraire. Substitute the exact value for m that I posted and LHS=RHS. This is elementary stuff.

@superacademy247 8 күн бұрын

WolframAlpha Calculator has some limitations. It doesn't have the capability of solving equations of this form. The solution in the video is 💯 correct. Thanks for watching and I appreciate your support.👌😎💡💕🤩✅

@Vega1447 8 күн бұрын

m=-log(r)/log(9/4) where r=log 4/ log 9.