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Chapter 12: Problem 6

Write each equation in its equivalent exponential form. $$3=\log _{b} 27$$

Short Answer

Expert verified
The equivalent exponential form of the given logarithmic equation \(3 = \log _{b} 27\) is \(b^3 = 27\).

Step by step solution

01

Identify the components of the logarithmic equation

In this equation \(3 = \log _{b} 27\), 3 is the logarithmic result (n), b is the base of the logarithm, and 27 is the argument of the logarithm (a). Keep in mind the structure of a logarithmic equation, \(\log _{b} a = n\).
02

Convert to exponential form

According to the equivalent form \(b^n = a\), we substitute the identified parts from the logarithmic equation into the exponential form. Here, b is the base, n corresponds to the result of the logarithm (which is 3), and a corresponds to the argument of the logarithm (which is 27). Hence, the equivalent exponential equation is \(b^3 = 27\).

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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 \(13 ?\)

Logarithmic models are well suited to phenomena in which growth is initially rapid, but then begins to level off. Describe something that is changing over time that can be modeled using a logarithmic function.

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

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{\log _{2} 8}{\log _{2} 4}=\frac{8}{4}$$

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$\log (x-15)+\log x=2$$
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