To simplify the given expression [tex]\(\log_b 4 + 2 \log_b 4\)[/tex] using logarithm properties, follow these steps:
1. Identify the logarithm properties: Recall the properties of logarithms, specifically the power rule and the product rule.
- The power rule: [tex]\(a \log_b x = \log_b(x^a)\)[/tex]
- The product rule: [tex]\(\log_b x + \log_b y = \log_b(x \cdot y)\)[/tex]
2. Apply the power rule: Rewrite [tex]\(2 \log_b 4\)[/tex] using the power rule.
[tex]\[
2 \log_b 4 = \log_b (4^2) = \log_b 16
\][/tex]
3. Combine the logarithms: Now, add [tex]\(\log_b 4\)[/tex] and [tex]\(\log_b 16\)[/tex] using the product rule.
[tex]\[
\log_b 4 + \log_b 16 = \log_b (4 \cdot 16)
\][/tex]
4. Multiply inside the logarithm:
[tex]\[
4 \cdot 16 = 64
\][/tex]
5. Write the final logarithmic expression: The simplified form of the original expression is:
[tex]\[
\log_b (64)
\][/tex]
So, the answer is:
[tex]\[
\log_b 64
\][/tex]
Therefore, [tex]\(\log_b 4 + 2 \log_b 4\)[/tex] can be written as:
[tex]\[
\log_b 64
\][/tex]
