Question

evaluate logarithms (level 3) unique id: 0062
score: 1/4 penalty: 0.25 off
question
evaluate:
\log_{128} 2

Explanation:

Step1: Recall logarithm definition

Let \( \log_{128} 2 = x \). By definition of logarithm, this means \( 128^x = 2 \).

Step2: Express 128 as power of 2

We know that \( 128 = 2^7 \), so substitute into the equation: \( (2^7)^x = 2 \).

Step3: Simplify left side

Using exponent rule \( (a^m)^n = a^{mn} \), we get \( 2^{7x} = 2^1 \).

Step4: Equate exponents

Since the bases are equal, the exponents must be equal: \( 7x = 1 \).

Step5: Solve for x

Divide both sides by 7: \( x = \frac{1}{7} \).

Answer:

\(\frac{1}{7}\)