Express $\log b \left( \frac { x ^ { \frac { 1 } { 2 } } y ^ { \frac { 1 } { 5 } } } { z ^ { 8 } } \right)$ as the sum or difference of simpler logarithmic quantities. Assume that all variables represent positive real numbers.
A) $\frac { 1 } { 2 } \log b x + \frac { 1 } { 5 } \log b y - 8 \log b z$
B) $\frac { 1 } { 8 } \log _ { b } x - 2 \log _ { b } y + \frac { 1 } { 5 } \log _ { b } z$
C) $8 \log _ { b } z - \frac { 1 } { 2 } \log _ { b } x + \frac { 1 } { 5 } \log b y$
D) $\frac { 1 } { 5 } \log b x + \frac { 1 } { 2 } \log b y - 8 \log b z$

Question

Quizplus · Accepted Answer

Using the properties of logarithms, the expression can be broken down into the sum and difference of simpler logarithmic quantities. The properties used are: $\log_b(mn) = \log_b m + \log_b n$ for multiplication, $\log_b\left(\frac{m}{n}\right) = \log_b m - \log_b n$ for division, and $\log_b(m^n) = n \log_b m$ for exponents. Applying these properties to the given expression results in $\frac{1}{2}\log_b x + \frac{1}{5}\log_b y - 8\log_b z$.

Express log⁡b(x12y15z8)\log b \left( \frac { x ^ { \frac { 1 } { 2 } } y ^ { \frac { 1 } { 5 } } } { z ^ { 8 } } \right)logb(z8x21​y51​​)

Express $\log b \left( \frac { x ^ { \frac { 1 } { 2 } } y ^ { \frac { 1 } { 5 } } } { z ^ { 8 } } \right)$