Express the following as a single logarithm. (Assume that all variables represent positive real numbers.) $3 \log b x + \frac { 1 } { 2 } \log _ { b } ( x - 2 ) - 7 \log b ( 3 x + 8 )$
A) $\log b \left( \frac { x ^ { 3 } \sqrt { x - 2 } } { ( 3 x + 8 ) ^ { 7 } } \right)$
B) $\log b \left( \frac { ( x - 2 ) ^ { 7 } \sqrt { x } } { ( 3 x + 8 ) ^ { 3 } } \right)$
C) $\log b \left( \frac { x ^ { 7 } } { \sqrt { x - 2 } ( 3 x + 8 ) ^ { 3 } } \right)$
D) $\log b \left( \frac { x ^ { 7 } } { \sqrt { 3 x + 8 } ( x - 2 ) ^ { 3 } } \right)$

Question

Quizplus · Accepted Answer

The expression can be simplified using logarithm rules: $3\log_b(x) = \log_b(x^3)$, $\frac{1}{2}\log_b(x-2) = \log_b(\sqrt{x-2})$, and $7\log_b(3x+8) = \log_b((3x+8)^7)$. Combining these using the rule $\log_b(a) - \log_b(c) = \log_b(\frac{a}{c})$ gives $\log_b \left( \frac { x ^ { 3 } \sqrt { x - 2 } } { ( 3 x + 8 ) ^ { 7 } } \right)$.

Express the Following as a Single Logarithm 3log⁡bx+12log⁡b(x−2)−7log⁡b(3x+8)3 \log b x + \frac { 1 } { 2 } \log _ { b } ( x - 2 ) - 7 \log b ( 3 x + 8 )3logbx+21​logb​(x−2)−7logb(3x+8)

Express the Following as a Single Logarithm $3 \log b x + \frac { 1 } { 2 } \log _ { b } ( x - 2 ) - 7 \log b ( 3 x + 8 )$