Problem 27
Question
Evaluate each expression without using a calculator. $$\log _{2} \frac{1}{8}$$
Step-by-Step Solution
Verified Answer
-3
1Step 1: Understanding the Problem
Recall that if \( a^b = c \), then we can rewrite this in logarithmic form as \( \log_a c = b \). Essentially, when the base 'a' is raised to the power 'b', it results in 'c'. Your task is to understand this relationship well in order to solve the problem.
2Step 2: Rewrite the Argument as a Power of 2
The base of the logarithm in the problem is 2, and the argument (the number you are taking the log of) is 1/8. Rewrite 1/8 as a power of 2. 1/8 can be rewritten as \( 2^{-3} \).
3Step 3: Apply the Logarithm Rule
Since 1/8 has been rewritten as \( 2^{-3} \), the logarithmic equation in this problem can be rewritten as \( \log_2 2^{-3} \). Recognize that \( \log_a a^b = b \), generally. Therefore, \( \log_2 2^{-3} = -3 \).
Other exercises in this chapter
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