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Chapter 12: Problem 21

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

Short Answer

Expert verified
The answer is \(2\).

Step by step solution

01

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.
02

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 \(4^x = 16\).
03

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 \(4^2 = 16\).

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When graphing a logarithmic function, I like to show the graph of its horizontal asymptote.

Because the equations \(2^{x}=15\) and \(2^{x}=16\) are similar, I solved them using the same method.

Without showing the details, explain how to condense \(\ln x-2 \ln (x+1)\)

Find \(\ln 2\) using a calculator. Then calculate each of the following: \(1-\frac{1}{2} ; \quad 1-\frac{1}{2}+\frac{1}{3} ; \quad 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}\) \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}, \ldots\) Describe what you observe.

a. Evaluate: \(\log _{2} 16\) b. Evaluate: \(\log _{2} 32-\log _{2} 2\) c. What can you conclude about $$\log _{2} 16, \text { or } \log _{2}\left(\frac{32}{2}\right) ?$$
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