Different bases, different logs and what? The unknown logged in the power?

Algebra Level 4

If $3^{\log (3x)} = 4^{\log (4x)}$ and $x=\frac{a}{b}$ where $a,b$ are coprime positive integers, what is the value of $a+b$ ?

The answer is 13.

1 solution

Noel Lo
Apr 20, 2015

$3^{log (3x)} = 4^{log (4x)}$

$log 3^{log (3x)} = log 4^{log (4x)}$

$(log 3x)(log 3) = (log 4x)(log 4)$

$(log 3+ log x)(log 3) = (log 4+ log x)(log 4)$

$(log 3)^2 + (log 3)(log x) = (log 4)^2 + (log 4)(log x)$

$(log 3)^2 - (log 4)^2 = (log 4)(log x) - (log 3)(log x)$

$(log 3 - log 4)(log 3+ log 4) = (log 4 - log 3)(log x)$

$-(log 4-log 3)(log 3+log 4) = (log 4- log 3)(log x)$

$-log 12 = log x$

$log \frac{1}{12} = log x$

$x=\frac{1}{12}$ so $a+b = 1+12 = \boxed{13}$ . :D

Similar method

Chew-Seong Cheong - 6 years, 1 month ago

Haha!! Hope you enjoyed this problem!!! :)

Noel Lo - 6 years, 1 month ago

I was thinking to provide solution, but you have done it. Anyway, I have tried your other problem . Your answer may be wrong. I believe $hg(i)$ should be $h(g(i))$ and $f^2(2) = f(f(2))$ , right? I got the answer to be $-21$ .

Chew-Seong Cheong - 6 years, 1 month ago

@Chew-Seong Cheong – Yes, you are right in your definition of the functions!!! I am very sorry but there is a mistake in the question! It should be -2 and NOT 2!! Really sorry once again for time lost in re-checking solutions!! :P Your answer should be -3.

Noel Lo - 6 years, 1 month ago

Different bases, different logs and what? The unknown logged in the power?

The answer is 13.

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