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Chapter 3: Problem 26

Use the One-to-One Property to solve the equation for \(x .\) $$5^{x-2}=\frac{1}{125}$$

Short Answer

Expert verified
The solution for \(x\) in the equation \(5^{x-2}=\frac{1}{125}\) is \(x = -1\).

Step by step solution

01

Rewriting the Equation

Rewrite the equation \(5^{x-2}=\frac{1}{125}\) in such a way that both sides have the same base. The number 125 can be expressed as \(5^3\), but because it is on the denominator, we can rewrite it as \(5^{-3}\). So, our equation becomes \(5^{x-2} = 5^{-3}\).
02

Applying the One-to-One Property

The One-to-One property of exponential functions states that if \(a^b = a^c\) (where \(a > 0\) and \(a \neq 1\)), then \(b = c\). Our equation \(5^{x-2} = 5^{-3}\) is in this form. Using this property, we simplify to \(x - 2 = -3\).
03

Solving for x

Now, simply solve the equation \(x -2 = -3\) for \(x\) by adding 2 to both sides of the equation to get \(x = -3 + 2 = -1\).

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Most popular questions from this chapter

Function \(\quad\) Value \(\begin{array}{ll}\text { 57. } f(x)=\ln x & x=18.42 \\ \text { 58. } f(x)=3 \ln x & x=0.74\end{array}\) \(\begin{array}{lll}y & f(x)=3 \ln x & x=0.74 \\ \text { 59. } g(x)=8 \ln x & x=0.05\end{array}\) 60\. \(g(x)=-\ln x \quad x=\frac{1}{2}\)

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