Problem 136 - College Algebra Essentials

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

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 1: 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.

Step 2: 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 \(\frac{4}{3}\).

Step 3: 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.

Step 4: 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.