solution for this logarthmic equation

Question

Is this solution correct?

$5^x = 3^{(x-1)}$

$\log_5 (3^{(x-1)}) = x$

$(x-1)\log_5 (3) = x$

$x\log_5 (3) - \log_5 (3) = x$

$x\log_5 (3) - x = \log_5 (3)$

$x(-1 + \log_5(3)) = \log_5 (3)$

$x = \cfrac{\log_5 (3)}{-1\log_5 (3)}$

$x = -1$

Answer 1

It's very hard for me to understand what you wrote, but the solution is

$$5^x=3^{x-1}\implies x\log 5=(x-1)\log 3\implies x(\log 5-\log 3)=-\log 3\implies$$

$$x=-\frac{\log 3}{\log5-\log3}=\frac{\log 3}{\log 3-\log 5}$$

In the first line, we used the logarithmic property $\;\log x^n=n\,\log x\;$ , and all the above is true to any valid base of the logarithm.