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Chapter 10: Problem 7

Solve each equation. Approximate solutions to three decimal places. $$ 9^{-x+2}=13 $$

Short Answer

Expert verified
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Step by step solution

01

Rewrite the equation in exponential form

The given equation is Rewrite it by changing the exponential base to a logarithmic equation:
02

Apply the logarithm on both sides

Take the natural logarithm of both sides to help simplify the equation:
03

Use the logarithmic property

Utilize the property of logarithms where to move the term with x:
04

Solve for x

Isolate x by rearranging the equation the right-hand side: Finally, solve for x and approximate to three decimal places.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Logarithm
The natural logarithm, denoted as \(\text{ln}\), is a special type of logarithm with a base of the mathematical constant \(e\) (approximately 2.71828). When solving equations involving exponential terms, using \(\text{ln}\) can make the process simpler. For example, in the equation \(9^{-x+2}=13\), taking the natural logarithm on both sides helps in dealing with the exponent: \(\text{ln}(9^{-x+2})=\text{ln}(13)\). This allows us to leverage logarithmic properties to manipulate the equation.
Exponential Form
An exponential equation features the variable in the exponent, like \(9^{-x+2}=13\). To solve such equations, it's often useful to rewrite them in logarithmic form. Using the natural logarithm can simplify the exponentiation: converting \(9^{-x+2}=13\) into \(\text{ln}(9^{-x+2})=\text{ln}(13)\). This conversion is crucial because it allows us to deal directly with the exponent, making further algebraic manipulation possible.
Logarithmic Properties
Logarithmic properties are essential tools in solving logarithmic and exponential equations. One critical property is that \(\text{ln}(a^b) = b \cdot \text{ln}(a)\). This property allows us to bring the exponent down: \(\text{ln}(9^{-x+2})=\text{ln}(13)\) becomes \(-(x-2) \cdot \text{ln}(9)=\text{ln}(13)\). Using properties like these helps simplify equations, making it easier to isolate and solve for the variable.
Isolating Variables
Isolating the variable is often the final step in solving an equation. Once the logarithmic properties have simplified the exponent, you need to manipulate the equation to solve for \(x\). For instance, \(\text{ln}(9^{-x+2})=\text{ln}(13)\) simplifies to \(-(x-2) \cdot \text{ln}(9)=\text{ln}(13)\). Then, isolate \(x\): \(-(x-2) = \frac{\text{ln}(13)}{\text{ln}(9)}\). Finally, solve for \(x\) by expanding and approximating to get the final answer.

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