Rewrite as a single logarithm. Assume all variables represent positive real numbers.
-$4 \log _ { 4 } ( 4 x + 2 ) + 3 \log _ { 4 } ( 5 x + 3 )$
A) $12 \log _ { 4 } ( 4 x + 2 ) ( 5 x + 3 )$
B) $\log _ { 4 } \frac { ( 4 x + 2 ) ^ { 4 } } { ( 5 x + 3 ) ^ { 3 } }$
C) $\log _ { 4 } \left( ( 4 x + 2 ) ^ { 4 } ( 5 x + 3 ) ^ { 3 } \right)$
D) $\log _ { 4 } \left( ( 4 x + 2 ) ^ { 4 } + ( 5 x + 3 ) ^ { 3 } \right)$

Question

Quizplus · Accepted Answer

The properties of logarithms allow us to rewrite the expression as a single logarithm by using the power rule, $\log_b(m^n) = n \log_b(m)$, and the product rule, $\log_b(m) + \log_b(n) = \log_b(mn)$. Applying these rules, we combine the terms into one logarithm with a base of 4, where the argument is the product of $(4x + 2)^4$ and $(5x + 3)^3$.

Rewrite as a Single Logarithm 4log⁡4(4x+2)+3log⁡4(5x+3)4 \log _ { 4 } ( 4 x + 2 ) + 3 \log _ { 4 } ( 5 x + 3 )4log4​(4x+2)+3log4​(5x+3)

Rewrite as a Single Logarithm $4 \log _ { 4 } ( 4 x + 2 ) + 3 \log _ { 4 } ( 5 x + 3 )$