How to solve $6^{2x} = 5^{x-2}$ for $x$

Question

I have recently come across this problem in my homework and I'm finding it quite difficult to solve. The simplest answer would be preferred using exponential laws and logarithms.

Answer 1

\begin{align}6^{2x}&=5^{x-2}\\ \log\left(6^{2x}\right)&=\log\left(5^{x-2}\right)\\ 2x\log(6)&=(x-2)\log(5)\tag{$\dagger$}\\ 2x\log(6)&=x\log(5)-2\log(5)\\ x\log(5)-2x\log(6)&=2\log(5)\\ x\left(\log(5)-2\log(6)\right)&=2\log(5)\\ x&=\frac{2\log(5)}{\log(5)-2\log(6)}\end{align}

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$(\dagger)$ We have used the rule that $\log\left(a^b\right)=b\log(a)$

Answer 2

$6^{2x} = 5^{x-2} \iff 36^x=5^x* \frac{1}{25} \iff (\frac{36}{5})^x=\frac{1}{25} $

logarithm ...........

Answer 3

Hint. By taking the logarithm of both sides we get the equivalent equation $$2x\ln(6)=\ln(6^{2x})=\ln(5^{x-2})=(x-2)\ln(5).$$ Are you able to find $x$ now?