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Chapter 12: Problem 18

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$9^{x}=\frac{1}{\sqrt[3]{3}}$$

Short Answer

Expert verified
The solution to the equation \(9^{x}=\frac{1}{\sqrt[3]{3}}\) is \(x=-1/6\).

Step by step solution

01

Express Each Side as a Power of the Same Base

Express both the left and right side of the equation as powers of the base number 3. The base number 3 was chosen because it is the common base for both 9 and 1/3. So, \(9^{x}\) can be written as \((3^{2})^{x}\) and \(\frac{1}{\sqrt[3]{3}}\) can be written as \(3^{-1/3}\)
02

Using Laws of Exponents Simplify the Bases

Simplify the left side of the equation using the property of exponents that says \((a^{b})^{c} = a^{bc}\). So, the left side of the equation simplifies to \(3^{2x}\). Hence, our equation is now \(3^{2x}=3^{-1/3}\)
03

Equate the Exponents

Since we have expressed both sides of the equation as powers of the same base (3), we can equate the exponents to each other. Therefore, setting the exponents equal to each other gives us \(2x=-1/3\)
04

Solve for \(x\)

To isolate \(x\), divide both sides of the equation by 2: \(x=-1/6\)

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Most popular questions from this chapter

Explain why the logarithm of 1 with base \(b\) is \(0 .\)

Explain how to find the domain of a logarithmic function.

a. Use a graphing utility (and the change-of-base property) to graph \(y=\log _{3} x\) b. Graph \(\quad y=2+\log _{3} x, \quad y=\log _{3}(x+2), \quad\) and \(y=-\log _{3} x \quad\) in the same viewing rectangle as \(y=\log _{3} x .\) Then describe the change or changes that need to be made to the graph of \(y=\log _{3} x\) to obtain each of these three graphs.

Without using a calculator, find the exact value of \(\log _{4}\left[\log _{3}\left(\log _{2} 8\right)\right]\)

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. $$5^{x}=3 x+4$$
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