Problem 29 - College Algebra

Question

Evaluate each expression without using a calculator. $$\log _{64} 8$$

Step-by-Step Solution

Answer

The value of \(log_{64} 8\) is \(1/2\).

Step 1 - Rewrite the given expression using change of base

The change of base formula in logarithms is useful here because there are common bases that can be used with the numbers 64 and 8. Specifically, both can be written as powers of 2. Therefore, let's use the change of base formula. Now, instead of \(log_{64} 8\), we can rewrite it using the base 2: \(log_{64} 8 = log_{2} 8 / log_{2} 64\)

Step 2 - Calculate the new logarithms

Now we calculate the values of these two new logarithms. \(log_{2} 8\) is asking 2 to the power of what equals 8? The answer is 3 because \(2^3 = 8\). Similar way for \(log_{2} 64\), 2 to the power of what equals 64? The answer is 6 because \(2^6 = 64\). Our new expression is now 3/6.

Step 3 - Simplify the result

The result of our new expression, 3/6, can be simplified to 1/2, which is the final answer.