Problem 21 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each expression without using a calculator. $$\log _{4} 16$$

Step-by-Step Solution

Answer

The answer is \(2\).

Step 1: Understanding logarithmic form

A logarithm with a certain base can be seen as asking what power or exponent we must raise the base to obtain the number. For this particular exercise, we have a base of 4, and the number is 16. The question is, to what power must we raise 4 to obtain 16.

Step 2: Convert Logarithm to Exponential Form

We may convert this from logarithmic form to exponential form. The base of the logarithm becomes the base of the exponent, the result of the logarithm becomes the exponent, and the content of the logarithm (e.g., what follows after the log) is the result of the exponentiation. Thus, we rewrite the logarithmic equation \(\log_{4} 16\) in exponential form as \(4^x = 16\).

Step 3: Solve the Exponential Equation

We now have an equation that asks to what power we must raise 4 in order to get 16. Solving this equation, we find that \(x = 2\), since \(4^2 = 16\).