Question

Use the properties of logarithms to expand the expression. Show your work by writing on paper or using the mathtype (Σ) option.

log(a5b)

Answer 1

To expand the logarithmic expression \( \log(a^5 b) \) using the properties of logarithms, we can apply the following properties:

1. **Product Rule**: \(\log(xy) = \log(x) + \log(y)\)
2. **Power Rule**: \(\log(x^n) = n \cdot \log(x)\)

Now, let's apply these rules step by step:

1. Start with the expression:
\[
\log(a^5 b)
\]

2. Apply the Product Rule:
\[
\log(a^5 b) = \log(a^5) + \log(b)
\]

3. Now, apply the Power Rule to \( \log(a^5) \):
\[
\log(a^5) = 5 \cdot \log(a)
\]

4. Substitute this back into the expression:
\[
\log(a^5 b) = 5 \cdot \log(a) + \log(b)
\]

So, the expanded form of \( \log(a^5 b) \) is:
\[
\boxed{5 \cdot \log(a) + \log(b)}
\]