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Chapter 4: Problem 6

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$3^{2 x+1}=27$$

Short Answer

Expert verified
The solution to the equation is \(x = 1\).

Step by step solution

01

Express both sides as powers of the same base

The given equation is \(3^{2 x+1}=27\). Since 27 can be written as \(3^3\), the equation can be rewritten as \(3^{2x+1} = 3^3\).
02

Equate Exponents

Since the bases on both sides of the equation are same (base 3), the exponents can be equated. Therefore, the equation becomes: \(2x + 1 = 3\).
03

Solve for x

Finally, solve the equation for x. Subtract 1 from both sides of the equation to get \(2x = 3 - 1 = 2\). Then divide both sides by 2 to isolate x: \(x = 2/2 = 1\).

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Most popular questions from this chapter

The data can be modeled by the function \(f(x)=1.2 \ln x+15.7\) where \(f(x)\) is the percentage of the U.S. gross domestic product going toward health care \(x\) years after \(2006 .\) a. Use the function to determine the percentage of the U.S. gross domestic product that went toward health care in \(2009 .\) Round to the nearest tenth of a percent. Does this underestimate or overestimate the percent displayed by the graph? By how much? b. According to the model, when will \(18.5 \%\) of the U.S. gross domestic product go toward health care? Round to the nearest year.

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. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age \(13 ?\)

Complete the table for a savings account subject to contimuous compounding ( \(A=P e^{n}\) ). Round answers to one decimal place. Amount Invested 8000 dollar Annual Interest Rate 20.3% Accumulated Amount 12,000 dollar Time \(t\) in Years _______

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. $$\log (x-15)+\log x=2$$

Explain how to use the graph of \(f(x)=2^{x}\) to obtain the graph of \(g(x)=\log _{2} x\).
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