Problem 136 - College Algebra

Question

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 \(8^{2}=64\).

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.