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Chapter 11: Problem 100

Solve: \(\log _{3}\left(\frac{x}{2}-1\right)=4\)

Short Answer

Expert verified
x = 164

Step by step solution

01

Understand the Problem

We need to solve the equation \(\log_{3}\left(\frac{x}{2}-1\right)=4.\)This means we need to find the value of \(x\) that satisfies this equation.
02

Exponentiate Both Sides

To remove the logarithm, rewrite the equation in its exponential form. Since \(3^{4} = 81\), the equation becomes: \(\frac{x}{2} - 1 = 81.\)
03

Solve for \(\frac{x}{2}\)

Add 1 to both sides of the equation: \(\frac{x}{2} = 81 + 1 \), which simplifies to \(\frac{x}{2} = 82.\)
04

Solve for \(x\)

Multiply both sides of the equation by 2 to isolate \(x\): \(x = 82 \times 2 \), which simplifies to \(x = 164.\)

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Key Concepts

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

Exponential Form
To solve logarithmic equations, one useful method is to convert them to their exponential form. This leverages the relationship between logarithms and exponents.
Logarithmic equations often look like \(\log_{b}(a)=c\), where \(b\) is the base of the logarithm, \(a\) is the argument, and \(c\) is the result of the logarithm.
In exponential form, this converts to \(b^{c}=a\).
For the given problem \(\log_{3}(\frac{x}{2}-1)=4\), converting it to exponential form gives \(3^{4}=\frac{x}{2}-1\).
Exponential form can make complicated equations simpler to solve by removing the logarithm and leaving a standard algebraic equation that is often easier to manage.
Solving Equations
Once you convert the logarithmic equation into exponential form, solving the equation becomes straightforward. Let's illustrate this with our solved problem.
The equation \(\frac{x}{2}-1=81\) was derived from the conversion step.
The steps to solve this equation break down into simple arithmetic:
  • Add 1 to both sides: \(\frac{x}{2}=81+1=82\)
  • Multiply both sides by 2 to solve for \(x\): \(x=82\times2=164\)
These steps systematically isolate the variable \(x\), helping us find the value that satisfies the original equation.
Logarithms
Understanding logarithms is essential for solving logarithmic equations. A logarithm is the inverse function of an exponentiation. If \(b^{c}=a\), then \((\log_{b}(a)=c)\).
In simple terms, a logarithm answers the question: 'To what power must the base be raised, to produce a given number?'
For example, in \(\log_{3}(81)=4\), it translates to asking: 'To what power must 3 be raised to get 81?' The answer is 4, because \(3^{4}=81\).
This inverse relationship is what allows us to convert between logarithmic form and exponential form, making problems easier to solve.

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