Logarithm Proof

I have been getting conflicting answers from others about what I thought was a pretty easy proof:

If $2^a = 3^{a+1}$ , show that $a = \frac{\log 3}{\log (2/3)}$ .

Any answers?

#Algebra

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

$\begin{aligned}2^a&=3^{a+1}\\a&=\log_2 3^{a+1}\\&=a\log_2 3+log_2 3\\a(1-\log_2 3)&=\log_2 3\\a&=\frac{\log_2 3}{1-\log_2 3}\\&=\frac{\frac{\log 3}{\log 2}}{1-\frac{\log 3}{\log 2}}\ (*)\\&=\frac{\log 3}{\log 2-\log 3}\\&=\color{#20A900}\boxed{\frac{\log 3}{\log \frac{2}{3}}}\ \color{#333333} \square\end{aligned}$

Proof of change of base formula ( $*$ ): $\begin{aligned}\text{Let }x&=\log_a b\\a^x&=b\\\log a^x&=\log b\\x\log a&=\log b\\x&=\frac{\log b}{\log a}\ \square\end{aligned}$

Matthew Christopher - 9 months, 3 weeks ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$