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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
The statement makes sense. Estimating that \(\log _{8} 16\) lies between 1 and 2 based on the powers of 8 is correct and fits with the properties of logarithms. The calculation of \(\log_{8}16\) confirms this as it is indeed in the range of 1 to 2.

Step by step solution

01

Understand the basics of logarithms

A logarithm is a way to solve for x in an exponential equation. The expression \(\log _{8} 16\) reads 'logarithm base 8 of 16'. Which means we are trying to find what power we need to raise 8 to, in order to get 16.
02

Estimation based on logarithm properties

The statement suggests that the value of \(\log _{8} 16\) would lie between 1 and 2 because \(8^{1}=8\) and \(8^{2}=64\). Since 16 lies between 8 and 64, this estimation seems reasonable. We are trying to find a value which when raised to the power of 8 gives 16, which is indeed between 8 and 64.
03

Verification

To confirm though, we can compute the actual value of \(\log_{8}16\) which is 1.5. This is indeed between 1 and 2 which confirms the reasoning provided in the statement is accurate.

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