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Chapter 12: Problem 97

Describe the power rule for logarithms and give an example.

Short Answer

Expert verified
The power rule for logarithms states that \( \log_b{a^n} = n \cdot \log_b{a} \). As for the instance \( \log_2{8^3} \), using the power rule, the solution will be 9.

Step by step solution

01

Describing the Power Rule

The power rule for logarithms states that the logarithm of a number raised to a certain power is equal to the power times the logarithm of the number itself. Mathematically, this concept is expressed as follows: \( \log_b{a^n} = n \cdot \log_b{a} \). Here, 'b' is the base of the logarithm, 'a' is the argument of the logarithm, and 'n' is the power to which 'a' is raised.
02

Giving an Example

Consider an example to illustrate this rule: \( \log_2{8^3} \). According to the power rule, this can be rewritten as: \( 3 \cdot \log_2{8} \). Now, we know that \( \log_2{8} = 3 \), so substituting this we get: \( 3 \cdot 3 = 9 \)

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Most popular questions from this chapter

The percentage of adult height attained by a girl who is \(x\) years old can be modeled by $$f(x)=62+35 \log (x-4)$$ where \(x\) represents the girl's age (from 5 to 15 ) and \(f(x)\) represents the percentage of her adult height. Use the formula to solve Exercises. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age \(13 ?\)

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 \(\log _{2}(8 \cdot 4) ?\)

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The percentage of adult height attained by a girl who is \(x\) years old can be modeled by $$f(x)=62+35 \log (x-4)$$ where \(x\) represents the girl's age (from 5 to 15 ) and \(f(x)\) represents the percentage of her adult height. Use the formula to solve Exercises. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age ten?
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