Question

finding inverse of exponential functions 1.) y=3^1/x2.) y= 10^x/3how exactly do I solve these problems

Answer 1

Firts, lets switch x and y

$\begin{gathered} y=3^{\frac{1}{x}}\rightarrow x=3^{\frac{1}{y}} \\ y=10^{\frac{x}{3}}\rightarrow x=10^{\frac{y}{3}} \end{gathered}$

In order to find the inverse, we have to clear y.

To do this with an exponential function, we use a little trick:

Apply log to both sides:

$\begin{gathered} x=3^{\frac{1}{y}}\rightarrow\log (x)=\log (3^{\frac{1}{y}}) \\ x=10^{\frac{y}{3}}\rightarrow\log (x)=\log (10^{\frac{y}{3}}) \end{gathered}$

We do this in order to use the property of logarithms that allow us to turn an exponent into a coefficient.

Mathematically speaking,

$\log (x^y)=y\log (x)$

(Note: Remember that log refers to natural logarithm; a logarithm with base e)

Thus,

$\begin{gathered} \log (x)=\frac{1}{y}\log (3) \\ \log (x)=\frac{y}{3}\log (10) \end{gathered}$

Clearing y to get the inverse,

$\begin{gathered} y=\frac{\log (3)}{\log (x)} \\ y=\frac{3\log (x)}{\log (10)} \end{gathered}$

Therefore,

$\begin{gathered} f^{-1}(x)=\frac{\log (3)}{\log (x)} \\ f^{-1}(x)=\frac{3\log (x)}{\log (10)} \end{gathered}$