Assume that x, y, z and b are positive numbers.Use the properties of logarithms to write the expression $\log _ { b } \sqrt [ 4 ] { \frac { x ^ { 5 } y ^ { 2 } } { z ^ { 4 } } }$ in terms of the logarithms of x, y, and z.
A) $\frac { 5 } { 4 } \log _ { b } x + \log _ { b } y - \log _ { b } z$
B) $\frac { 5 } { 4 } \log _ { b } x + \frac { 1 } { 2 } \log _ { b } y - \log _ { b } z$
C) $\log _ { b } x + \frac { 1 } { 2 } \log _ { b } y - \log _ { b } z$
D) $\frac { 5 } { 4 } \log _ { b } ( x + y + z )$
E) $20 \log _ { b } x + 8 \log _ { b } y - 16 \log _ { b } z$

Question

Assume that x, y, z and b are positive numbers.Use the properties of logarithms to write the expression $\log _ { b } \sqrt [ 4 ] { \frac { x ^ { 5 } y ^ { 2 } } { z ^ { 4 } } }$ in terms of the logarithms of x, y, and z. 
A) $\frac { 5 } { 4 } \log _ { b } x + \log _ { b } y - \log _ { b } z$
B) $\frac { 5 } { 4 } \log _ { b } x + \frac { 1 } { 2 } \log _ { b } y - \log _ { b } z$
C) $\log _ { b } x + \frac { 1 } { 2 } \log _ { b } y - \log _ { b } z$
D) $\frac { 5 } { 4 } \log _ { b } ( x + y + z )$
E) $20 \log _ { b } x + 8 \log _ { b } y - 16 \log _ { b } z$

Quizplus · Accepted Answer

The expression can be simplified using logarithm properties: $\log_b \sqrt[4]{\frac{x^5y^2}{z^4}}$ can be rewritten as $\frac{1}{4}\log_b \left(\frac{x^5y^2}{z^4}\right)$, which further simplifies to $\frac{1}{4}(5\log_b x + 2\log_b y - 4\log_b z)$ using the properties of logarithms for multiplication and division inside the log function. Simplifying this gives $\frac{5}{4}\log_b x + \frac{1}{2}\log_b y - \log_b z$.

Assume That X, Y, Z and B Are Positive Numbers log⁡bx5y2z44\log _ { b } \sqrt [ 4 ] { \frac { x ^ { 5 } y ^ { 2 } } { z ^ { 4 } } }logb​4z4x5y2​​

Assume That X, Y, Z and B Are Positive Numbers $\log _ { b } \sqrt [ 4 ] { \frac { x ^ { 5 } y ^ { 2 } } { z ^ { 4 } } }$