Expand. Assume that all variables represent positive real numbers.
-$\log _ { b } \left( \frac { x y ^ { 8 } } { z ^ { 7 } } \right)$
A) $\log _ { b } x + \log _ { b } y ^ { 8 } - \log _ { b } z ^ { 7 }$
B) $\log _ { \mathrm { b } } x + 8 \log _ { \mathrm { b } } \mathrm { y } - 7 \log _ { \mathrm { b } } z$
C) $\log _ { b } x + 8 \log _ { b } y + 7 \log _ { b } z$
D) $\log _ { b } x + \log _ { b } y ^ { 8 } + \log _ { b } z ^ { 7 }$

Question

Quizplus · Accepted Answer

By the properties of logarithms, we have:

$$\log_b\left(\frac{xy^8}{z^7}ight) = \log_b(x) + \log_b\left(y^8ight) - \log_b\left(z^7ight)$$

Then, using the power property of logarithms, we have:

$$\log_b(y^8) = 8\log_b(y)$$

$$\log_b(z^7) = 7\log_b(z)$$

Substituting these back into the original expression gives:

$$\log_b\left(\frac{xy^8}{z^7}ight) = \log_b(x) + 8\log_b(y) - 7\log_b(z)$$

Therefore, the correct choice is $\boxed{	ext{B}}$.

Expand Assume That All Variables Represent Positive Real Numbers log⁡b(xy8z7)\log _ { b } \left( \frac { x y ^ { 8 } } { z ^ { 7 } } \right)logb​(z7xy8​)

Expand Assume That All Variables Represent Positive Real Numbers $\log _ { b } \left( \frac { x y ^ { 8 } } { z ^ { 7 } } \right)$