Question

match each logarithm to its value. a. \\(\log_{27} 3\\) b. \\(\log_{81} 27\\) c. \\(\log_{9} 27\\) d. \\(\log_{\frac{1}{3}} 27\\) logarithm dropdown is equal to \\(-3\\). logarithm dropdown with a, b, c, d is equal to \\(\frac{3}{2}\\). logarithm dropdown with a, b, c, d is equal to \\(\frac{3}{4}\\). logarithm dropdown is equal to \\(\frac{1}{3}\\).

Explanation:

Step1: Rewrite logs with base 3

Recall that $27=3^3$, $81=3^4$, $9=3^2$, $\frac{1}{3}=3^{-1}$.

• A. $\log_{27}3 = \log_{3^3}3$
• B. $\log_{81}27 = \log_{3^4}3^3$
• C. $\log_{9}27 = \log_{3^2}3^3$
• D. $\log_{\frac{1}{3}}27 = \log_{3^{-1}}3^3$

Step2: Apply log power rule

Use $\log_{a^m}a^n = \frac{n}{m}$.

• A. $\log_{3^3}3 = \frac{1}{3}$
• B. $\log_{3^4}3^3 = \frac{3}{4}$
• C. $\log_{3^2}3^3 = \frac{3}{2}$
• D. $\log_{3^{-1}}3^3 = \frac{3}{-1} = -3$

Step3: Match values to logs

Pair each result with the given value.

Answer:

Logarithm D is equal to $-3$.
Logarithm C is equal to $\frac{3}{2}$.
Logarithm B is equal to $\frac{3}{4}$.
Logarithm A is equal to $\frac{1}{3}$.