Chapter 3.3, Problem 74E

Solution

To calculate: How to transform into graph.

It will reflect about $x-\text{axis}$ .

Given information: function $f\left(x\right)=\log x$ and $g\left(x\right)={\log }_{0.1}x$ .

Formula used: Standard form of logarithm to exponential form:

  $\begin{array}{l}y=\log x\\ x={10}^y\end{array}$

Calculation: In exponential form, $f(x)$

can be written as:

  $\begin{array}{l}y=\log x\\ x={10}^y\end{array}$

While $g\left(x\right)$ can be written as:

  $\begin{array}{c}y={\log }_{0.1}x\\ x={0.1}^y\\ ={\left(\frac{1}{10}\right)}^y\\ $ ={({10}^{-1})}^y$={10}^{-y}\end{array}$

This follows $g(x)=-f(x)$

Since sign of $y$ is changed, then graph of $f(x)$ can be transformed into graph of $g(x)$ by reflection across $x-\text{axis}$ .