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Chapter 4: Problem 107

Describe the power rule for logarithms and give an example.

Short Answer

Expert verified
The power rule for logarithms states that \(log_b (a^n) = n log_b (a)\). A practical example is \(log_3 (9^2) = 2 log_3 (9)\)

Step by step solution

01

Describe the Power Rule

The power rule of logarithms states that for any base \(b\), \(log_b (a^n) = n log_b (a)\). This means that when a number inside the log is raised to a power, we can take the power out and multiply it with the logarithm.
02

Provide an example using the Power Rule

Let's demonstrate how to apply the power rule using the logarithm \(log_3 (9^2) = 2 log_3 (9)\). The original logarithm has the number 9 raised to the power of 2. According to the Power Rule, we can take this power out and multiply it with the logarithm, resulting in \(2 log_3 (9)\).

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Most popular questions from this chapter

The figure shows the graph of \(f(x)=\ln x\). Use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. (GRAPH CANNOT COPY). $$ g(x)=1-\ln x $$

Find the domain of each logarithmic function. $$ f(x)=\ln \left(x^{2}-4 x-12\right) $$

Evaluate or simplify each expression without using a calculator. $$ 10^{\log 33} $$

Will help you prepare for the material covered in the next section. In each exercise, evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{2} 32\) b. Evaluate: \(\log _{2} 8+\log _{2} 4\) c. What can you conclude about \(\log _{2} 32,\) or \(\log _{2}(8 \cdot 4) ?\)

Will help you prepare for the material covered in the next section. In each exercise, evaluate the indicated logarithmic expressions without using a calculator. 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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