Question

question
evaluate:
\\(\log_{128} \frac{1}{16}\\)
answer attempt 1 out of 2

Explanation:

Step1: Express 128 and 16 as powers of 2

We know that \( 128 = 2^7 \) and \( 16 = 2^4 \), so \( \frac{1}{16}=2^{-4} \). Then the logarithm becomes \( \log_{2^7} 2^{-4} \).

Step2: Use the change of base formula for logarithms

The change of base formula for logarithms is \( \log_{a^m} b^n=\frac{n}{m}\log_a b \). When \( a = b \), \( \log_a a = 1 \). So \( \log_{2^7} 2^{-4}=\frac{-4}{7}\log_2 2 \).

Step3: Evaluate \( \log_2 2 \)

Since \( \log_a a = 1 \) for any positive \( a
eq1 \), \( \log_2 2 = 1 \). Then \( \frac{-4}{7}\times1=-\frac{4}{7} \).

Answer:

\( -\frac{4}{7} \)