Question

log power rule example

Answer 1

The logarithmic power rule states that the logarithm of a number raised to an exponent can be expressed as the exponent multiplied by the logarithm of that number. The rule can be formulated as follows:

\[
\log_b(a^n) = n \cdot \log_b(a)
\]

where:
- \( b \) is the base of the logarithm,
- \( a \) is the number being raised to the power,
- \( n \) is the exponent.

### Example 1:

Let's say we want to simplify \(\log_2(8^3)\).

1. Apply the power rule:
\[
\log_2(8^3) = 3 \cdot \log_2(8)
\]

2. Now calculate \(\log_2(8)\):
\[
8 = 2^3 \quad \Rightarrow \quad \log_2(8) = 3
\]

3. Substitute this back into the equation:
\[
3 \cdot \log_2(8) = 3 \cdot 3 = 9
\]

So, \(\log_2(8^3) = 9\).

### Example 2:

Now, consider a base of 10: simplify \(\log_{10}(1000^2)\).

1. Apply the power rule:
\[
\log_{10}(1000^2) = 2 \cdot \log_{10}(1000)
\]

2. Now calculate \(\log_{10}(1000)\):
\[
1000 = 10^3 \quad \Rightarrow \quad \log_{10}(1000) = 3
\]

3. Substitute this back:
\[
2 \cdot \log_{10}(1000) = 2 \cdot 3 = 6
\]

So, \(\log_{10}(1000^2) = 6\).

These examples illustrate how to use the logarithmic power rule effectively to simplify logarithmic expressions.