Answer
$$
\begin{aligned} \frac{d}{d x} \ln |a x| &=\frac{d}{d x}(\ln |a|+\ln |x|) \\ &=\frac{d}{d x} \ln |a|+\frac{d}{d x} \ln x \\
& \quad\quad\quad\left[\begin{array}{c}{ \text {since} \ln |a| \text { is a constant then} }\end{array}\right] \\
&=0+\frac{d}{d x} \ln |x| \\ &=\frac{d}{d x} \ln |x|. \end{aligned}
$$
Thus
$$
\frac{d}{d x} \ln |a x| =\frac{d}{d x} \ln |x| .
$$
Work Step by Step
$$
\begin{aligned} \frac{d}{d x} \ln |a x| &=\frac{d}{d x}(\ln |a|+\ln |x|) \\ &=\frac{d}{d x} \ln |a|+\frac{d}{d x} \ln x \\
& \quad\quad\quad\left[\begin{array}{c}{ \text {since} \ln |a| \text { is a constant then} }\end{array}\right] \\
&=0+\frac{d}{d x} \ln |x| \\ &=\frac{d}{d x} \ln |x|. \end{aligned}
$$
Thus
$$
\frac{d}{d x} \ln |a x| =\frac{d}{d x} \ln |x| .
$$
