Question

cphs : advanced algebra: concepts and connections - block (27.0831030)
properties of logarithms
what is \\(\log_{5}(4 \cdot 7) + \log_{5} 2\\) written as a single log?
\\(\log_{5} 30\\)
\\(\log_{5} 26\\)
\\(\log_{5} 56\\)
\\(\log_{5} 21\\)

Explanation:

Step1: Apply log product rule

The product rule of logarithms states that $\log_b(M) + \log_b(N) = \log_b(M \times N)$. Here, $b = 5$, $M = 4 \cdot 7 = 28$, and $N = 2$. So we first simplify the argument inside the logs: $4 \cdot 7 = 28$, then we have $\log_5(28) + \log_5(2)$.

Step2: Use the product rule

Applying the product rule $\log_5(28) + \log_5(2) = \log_5(28 \times 2)$. Calculate $28 \times 2 = 56$. So the expression becomes $\log_5(56)$.

Answer:

$\log_5 56$