Question

use the given function f to answer parts (a) through (f) below.

f(x)=\frac{1}{2}\log(5x)

(type your answer in interval notation.)

determine the vertical asymptote of f. select the correct choice below and, if necessary, fill in
box to complete your choice.

a. the vertical asymptote of f is x=
(simplify your answer.)

b. there is no vertical asymptote.

Explanation:

Step1: Recall the domain of the logarithmic function

For the function \(y = \log(u)\), the argument \(u>0\). In the function \(f(x)=\frac{1}{2}\log(5x)\), we set \(u = 5x\). So, \(5x>0\), which gives \(x>0\).

Step2: Determine the vertical asymptote

The vertical asymptote of a logarithmic function \(y=\log(ax + b)\) (where \(a
eq0\)) occurs at the value of \(x\) that makes the argument of the logarithm equal to \(0\). For \(y = \log(5x)\), when \(5x=0\) (i.e., \(x = 0\)), the function is undefined. As \(x\to0^{+}\), \(\log(5x)\to-\infty\).

Answer:

A. The vertical asymptote of \(f\) is \(x = 0\)