To find the value of \(\log_b(ab)\), let's break it down using the properties of logarithms.

### Step 1: Apply the Product Rule
The product rule for logarithms states:
\[
\log_b(xy) = \log_b(x) + \log_b(y)
\]
Here, \(x = a\) and \(y = b\). Therefore:
\[
\log_b(ab) = \log_b(a) + \log_b(b)
\]

### Step 2: Simplify \(\log_b(b)\)
By definition, \(\log_b(b) = 1\) because any logarithm of a number to its own base is 1. So:
\[
\log_b(ab) = \log_b(a) + 1
\]

### Conclusion
The value of \(\log_b(ab)\) is \(\log_b(a) + 1\).

Would you like to delve deeper into logarithm properties or have any further questions? Here are some follow-up questions you might consider:

1. How does the logarithm product rule work for different bases?
2. Can you derive the value of \(\log_b(a^2)\)?
3. How would you simplify \(\log_b(a^n)\)?
4. What is the relationship between \(\log_b(a)\) and \(\log_a(b)\)?
5. How can logarithms be used to solve exponential equations?

**Tip:** When working with logarithms, always remember to apply the fundamental rules: product, quotient, and power rules.

Understand how to calculate \( \log_b(ab) \) using logarithm properties. Learn step-by-step with the product rule applied, simplifying \( \log_b(b) \), and concluding with the value \( \log_b(a) + 1 \). Explore related logarithm rules and their applications.

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