$\log \mathrm{ab}-\log \left|\mathrm{b}\right|=$

In this question, we are tasked with simplifying the expression:

log ab - log |b|.

Let's break down the expression step by step:

Step 1: Understanding the Logarithmic Property

The logarithmic property we will apply here is the logarithmic rule for subtraction, which states that:

log x - log y = log (x/y)

Using this rule, we can rewrite the given expression as:

log ab - log |b| = log (ab / |b|)

Step 2: Simplifying the Expression

Now, let's simplify the fraction inside the logarithm:

ab / |b| = |a|

Here, we recognize that dividing ab by |b| leaves us with the absolute value of a, since |b| / b results in a positive or negative value of 1.

Step 3: Final Simplification

So, the expression becomes:

log (|a|)

This is the final simplified result. Therefore, we have:

log ab - log |b| = log |a|