Answer
$\log_5{\left(\dfrac{2x+2}{(3x+7)^{\frac{1}{3}}}\right)}$
Work Step by Step
RECALL:
(1) $n \cdot \log_a{P}=\log_a{(P^n)}$
(2) $\log_a{P} + \log_a{Q}=\log_a{(PQ)}$
(3) $\log_a{P} - \log_a{Q}=\log_a{(\frac{P}{Q})}$
(4) $a^m \cdot a^n=a^{m+n}$
Use rule (1) above to obtain
$\log_5{2}+\log_5{(x+1)}-\frac{1}{3}\log_5{(3x+7)}
\\=\log_5{2}+\log_5{(x+1)}-\log_5{(3x+7)^{\frac{1}{3}}}$
Use rule (2) above to obtain:
$\log_5{2}+\log_5{(x+1)}-\log_5{(3x+7)^{\frac{1}{3}}}
\\=\log_5{[2(x+1)]}-\log_5{(3x+7)^{\frac{1}{3}}}
\\=\log_5{(2x+2)}-\log_5{(3x+7)^{\frac{1}{3}}}$
Use rule (3) above to obtain:
$\log_5{(2x+2)}-\log_5{(3x+7)^{\frac{1}{3}}}
\\=\log_5{\left(\dfrac{2x+2}{(3x+7)^{\frac{1}{3}}}\right)}$
