Chapter 3.3, Problem 72E

Solution

To calculate: Why zero is not in the domain.

Reflected about $x-\text{axis}$ .

Given information: Function $f\left(x\right)={\log }_{3}x\text{ and g}\left(x\right)={\log }_{5}x$ .

Formula used: logarithm

Calculation: Here we use change of base property, by which

  ${\log }_{b}a=\frac{{\log }_{c}a}{{\log }_{c}b}$

We have seen this property in algebra and now review it in next section.

If we let $c=e$ is property as given below:

  ${\log }_{b}a=\frac{\ln a}{\ln b}$

So, we can rewrite $g\left(x\right)$ as

  $g(x)=\frac{\ln x}{\ln \frac{1}{e}}$

But of course,

  $\begin{array}{c}\ln \frac{1}{e}=\ln e^{-1}\\ =-1\end{array}$

And so,

  $g(x)=-\ln x$

Obtained reflecting graph of $f$ about the $x-\text{axis}$