Question

use properties of logarithms to condense the logarithmic expression. write the expression as a single logarithm whose coefficient is 1. evaluate logarithmic expressions if possible.
7\log_{b}x + 3\log_{b}z

7\log_{b}x + 3\log_{b}z = \square

Explanation:

Step1: Apply Power Rule

The power rule of logarithms states that \( n\log_b M = \log_b M^n \). Apply this to each term:

• For \( 7\log_b x \), we get \( \log_b x^7 \).
• For \( 3\log_b z \), we get \( \log_b z^3 \).

Step2: Apply Product Rule

The product rule of logarithms states that \( \log_b M + \log_b N = \log_b (M \cdot N) \). Combine the two logarithms from Step 1:
\( \log_b x^7 + \log_b z^3 = \log_b (x^7 \cdot z^3) \)

Answer:

\( \log_b (x^7 z^3) \)