Question

What is the inverse of the function f(x) = log (2x)?

Answer 1

as you already know, to get the inverse of any expression we start off by doing a quick switcheroo on the variables and then solving for "y", let's do so.

`\begin{array}{llll} \textit{Logarithm Cancellation Rules} \\\\ log_a a^x = x\qquad \qquad \stackrel{ \textit{let's use this one} }{a^(log_a x)=x} \end{array} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{f(x)}{y}~~ = ~~\log(2x)\hspace{5em}\stackrel{\textit{quick switcheroo}}{x~~ = ~~\log(2y)}\implies x~~ = ~~\log_(10)(2y) \\\\\\ 10^x~~ = ~~10^{\log_(10)(2y)}\implies 10^x=2y\implies \cfrac{10^x}{2}=y=f^(-1)(x)`

recall, when the logarithm base is omitted, "10" is implied.