Express $\log b \left( x \sqrt [ 6 ] { \frac { x } { y } } \right)$ as the sum or difference of simpler logarithmic quantities. Assume that all variables represent positive real numbers.
A) $\frac { 7 } { 6 } \log b y + \frac { 1 } { 6 } \log b x$
B) $\frac { 7 } { 6 } \log _ { b } x - \frac { 1 } { 6 } \log b y$
C) $\frac { 7 } { 6 } \log b y - \frac { 1 } { 6 } \log b x$

Question

Quizplus · Accepted Answer

Using the properties of logarithms, the expression can be simplified as follows: $\log_b(x \sqrt[6]{\frac{x}{y}}) = \log_b(x) + \log_b\left(\sqrt[6]{\frac{x}{y}}\right) = \log_b(x) + \frac{1}{6}\log_b\left(\frac{x}{y}\right) = \log_b(x) + \frac{1}{6}(\log_b(x) - \log_b(y)) = \frac{7}{6}\log_b(x) - \frac{1}{6}\log_b(y)$.

Express log⁡b(xxy6)\log b \left( x \sqrt [ 6 ] { \frac { x } { y } } \right)logb(x6yx​​)

Express $\log b \left( x \sqrt [ 6 ] { \frac { x } { y } } \right)$