Certainly! Let's tackle the problem step-by-step to convert the given logarithm into the sum of logarithms using the product property.
Given:
[tex]\[ \log_7[(x + y) \cdot z] \][/tex]
We need to express this logarithm as a sum of simpler logarithms. We'll use the product property of logarithms, which states:
[tex]\[ \log_b(M \cdot N) = \log_b(M) + \log_b(N) \][/tex]
Here, [tex]\( b = 7 \)[/tex], [tex]\( M = x + y \)[/tex], and [tex]\( N = z \)[/tex].
Applying the product property, we get:
[tex]\[ \log_7[(x + y) \cdot z] = \log_7(x + y) + \log_7(z) \][/tex]
Therefore, the logarithm expressed as a sum of logarithms is:
[tex]\[ \boxed{\log_7[(x + y) \cdot z] = \log_7(x + y) + \log_7(z)} \][/tex]
