Use the Laws of Logarithms to rewrite $\log _ { 11 } \sqrt [ 3 ] { 10 }$ in a form with no logarithms of products, quotients, or powers.
A) $\frac { 1 } { 11 } \log _ { 3 } 10$
B) $\log _ { 11 } 10 - \log _ { 11 } 3$
C) $\sqrt [ 3 ] { \log _ { 11 } 10 }$
D) $\frac { 1 } { 3 } \log _ { 11 } 10$
E) $\frac { 1 } { 3 } \log _ { 1 } 3$

Question

Quizplus · Accepted Answer

We can use the following laws of logarithms:

1. $\log_{a^n}b^n=\log_{a}b$
2. $\log_{a}(b\cdot c)=\log_{a}b+\log_{a}c$
3. $\log_{a}\frac{b}{c}=\log_{a}b-\log_{a}c$

Using these laws, we have:

$$\log _ { 11 } \sqrt [ 3 ] { 10 } = \frac{1}{3}\log_{11}10$$

Therefore, the correct answer is D.

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Use the Laws of Logarithms to Rewrite $\log _ { 11 } \sqrt [ 3 ] { 10 }$