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Chapter 4: Problem 136

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I estimate that \(\log _{8} 16\) lies between 1 and 2 because \(8^{1}=8\) and \(8^{2}=64\).

Short Answer

Expert verified
Yes, the statement makes sense because the evaluated logarithm \(\log _{8} 16\) equals \(\frac{4}{3}\) which lies between 1 and 2. The reasoning provided about \(8^{1}=8\) and \(8^{2}=64\) correctly explains that the logarithm of 16 should be between those two values.

Step by step solution

01

Understand the statement

The statement is about the logarithm function \(\log _{8} 16\). It states that this logarithmic expression lies between 1 and 2 because \(8^{1}=8\) and \(8^{2}=64\). The reasoning is based on the definition of the logarithm function, which is the inverse of the exponential function.
02

Evaluate the Logarithmic Expression

To evaluate the correctness of the statement, determine the exact value of \(\log _{8}16\). Using the base change rule of logarithms, \(\log _{8}16\) can be rewritten as \(\frac{\log _{2}16}{\log _{2}8}\). This evaluates to \(\frac{4}{3}\).
03

Compare the value

The evaluated logarithm, \(\frac{4}{3}\), roughly equals 1.33. This falls between 1 and 2, agreeing with the initial prediction in the statement.
04

Confirm the relationship

Revisiting the reason provided in the statement, \(8^1=8\) and \(8^2=64\), and knowing the definition of a logarithm as an inverse of exponentiation, it's clear that the value of \(\log _{8}16\) has to exist between 1 (as \(8^1<16\)) and 2 (as \(8^2>16\)). Hence the statement is sensible.

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