Question

What is the inverse function of f(x)=4^x

Please show the steps and try to explain the steps as well.

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 of exponentials} \\\\ \log_a\left( x^b \right)\implies b\cdot \log_a(x) \end{array} \\\\\\ \textit{Logarithm Change of Base Rule} \\\\ \log_a b\implies \cfrac{\log_c b}{\log_c a}\qquad \qquad c= \begin{array}{llll} \textit{common base for }\\ \textit{numerator and}\\ denominator \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}`

`\stackrel{f(x)}{y}~~ = ~~4^x\hspace{5em}\stackrel{\textit{quick switcheroo}}{x~~ = ~~4^y} \\\\\\ \log(x)=\log(4^y)\implies \log(x)=y\log(4)\implies \cfrac{\log(x)}{\log(4)}=y\implies \stackrel{ ~\hfill f^(-1)(x) }{\boxed{\log_4(x)=y}}`