Chapter 6.3, Problem 43E

To determine

To calculate: The inverse of the function.

Answer to Problem 43E

The inverse of the function $y={0.3}^x$ is ${\log }_{0.3}y=x$ .

Explanation of Solution

Given information:

The given function is $y={0.3}^x$ .

Formula Used:

Logarithmic power rule is

If $a^b$ is a function then logarithm form ${\log }_a{\left(a\right)}^b$ is equal to $b$ .

Calculation:

Consider the function is $y={0.3}^x$ .

Convert the exponential form into logarithmic form.

Take logarithm in each side. Use the logarithmic power rule.

  $\begin{array}{c}\log y=\log {0.3}^x\\ \log y=x\times \log 0.3\end{array}$

It is known that for two log function $\log a\text{and}\log b$ , their division $\frac{\log a}{\log b}={\log }_{b}a$ .

Now, solve the above expression, use the formula $\frac{\log a}{\log b}={\log }_{b}a$ .

  $\begin{array}{c}\log y=x\times \log 0.3\\ x=\frac{\log y}{\log 0.3}\\ ={\log }_{0.3}y\end{array}$

Hence, the inverse of the function $y={0.3}^x$ is ${\log }_{0.3}y=x$ .