Simplify the expression $\log _ { 3 } \left( \frac { 1 } { 27 } \right) ^ { 3 }$
A) 3
B) -9
C) 0
D) -81
E) The expression cannot be simplified.

Question

Quizplus · Accepted Answer

First, let's simplify the expression inside the parentheses: 
$$\left( \frac { 1 } { 27 } \right) ^ { 3 } = \frac{1}{27} \cdot \frac{1}{27} \cdot \frac{1}{27} = \frac{1}{27^3}$$
Now we can use the property $$\log_a (b^c) = c \cdot \log_a b$$
with $a=3$, $b=\frac{1}{27}$, and $c=3$ to get: 
$$\log_3 \left(\left(\frac{1}{27}\right)^3\right) = 3 \cdot \log_3 \left(\frac{1}{27}\right)$$
We can simplify $\log_3 \left(\frac{1}{27}\right)$ using the property $\log_a \left(\frac{1}{b}\right) = -\log_a b$ with $a=3$, $b=27$:
$$\log_3 \left(\frac{1}{27}\right) = -\log_3 27 = -\log_3 3^3 = -3$$
Substituting in the above expression gives:
$$3 \cdot \log_3 \left(\frac{1}{27}\right) = 3 \cdot (-3) = \boxed{\mathrm{(B)}\ -9}.$$

Simplify the Expression log⁡3(127)3\log _ { 3 } \left( \frac { 1 } { 27 } \right) ^ { 3 }log3​(271​)3

Simplify the Expression $\log _ { 3 } \left( \frac { 1 } { 27 } \right) ^ { 3 }$