Question

question
write the expression below as a single logarithm in simplest form.
$3\log_{b} 2 + \log_{b} 5$

Explanation:

Step1: Apply power rule of logarithms

The power rule states that \( n\log_b a=\log_b a^n \). So for \( 3\log_b 2 \), we can rewrite it as \( \log_b 2^3 \). Calculating \( 2^3 = 8 \), so \( 3\log_b 2=\log_b 8 \).

Step2: Apply product rule of logarithms

The product rule states that \( \log_b a+\log_b c = \log_b(ac) \). Now we have \( \log_b 8+\log_b 5 \), so applying the product rule, we get \( \log_b(8\times5) \).

Step3: Simplify the product

Calculating \( 8\times5 = 40 \), so the expression simplifies to \( \log_b 40 \).

Answer:

\(\log_{b} 40\)