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Chapter 10: Problem 65

Solve each equation. $$\log _{1 / 3}(x-4)=2$$

Short Answer

Expert verified
x = \frac{37}{9}

Step by step solution

01

Understand the Logarithmic Equation

To solve the equation \(\log _{1 / 3}(x-4)=2\), recall that a logarithmic equation \(\log_b (a) = c\) can be rewritten in its exponential form as \(b^c = a\). Here, \(b = \frac{1}{3}\), \(a = x - 4\), and \(c = 2\).
02

Rewrite the Logarithmic Equation in Exponential Form

Rewrite the given logarithmic equation \(\log_{1/3}(x-4) = 2\) into its exponential form. This gives: \(\left(\frac{1}{3}\right)^2 = x - 4\).
03

Simplify the Exponential Expression

Calculate \(\left(\frac{1}{3}\right)^2\). This is \(\frac{1}{3} \cdot \frac{1}{3} = \frac{1}{9}\). So, the equation now reads \(\frac{1}{9} = x - 4\).
04

Solve for x

To find the value of \(x\), add 4 to both sides of the equation: \(\frac{1}{9} + 4 = x\). Convert 4 into a fraction with the denominator 9: \(4 = \frac{36}{9}\). Now add these fractions: \(\frac{1}{9} + \frac{36}{9} = \frac{37}{9}\). Hence, \(x = \frac{37}{9}\).

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Key Concepts

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

Logarithmic Functions
Logarithmic functions are the inverse of exponential functions. That means if you have an exponential function like \( y = b^x \), the corresponding logarithmic function will be \( x = \log_b(y) \). Here, \( b \) is the base of the logarithm, and it must be a positive number not equal to 1. Logarithms help us solve for the exponent in exponential equations. For example, if you have \( \log_{1/3}(x-4) = 2 \), you can convert it to its exponential form \( (1/3)^2 = x-4 \).
Exponential Equations
Exponential equations are equations where variables appear as exponents. A common form of these equations is \( a * b^{cx} = d \). To solve exponential equations, you often need to use logarithms. For example, to solve the equation \( \left(\frac{1}{3}\right)^2 = x - 4 \), first recognize that \( \left(\frac{1}{3}\right)^2 \) equals \( \frac{1}{9} \). Then, the equation turns into \( \frac{1}{9} = x - 4 \). Use algebraic operations to isolate \( x \). Add 4 to both sides to find \( x \)
Solving Equations
Solving equations is the process of finding the value of the variables that make the equation true. For logarithmic and exponential equations, it often involves the following steps:
  • Rewrite the equation in a different form (logarithmic or exponential).
  • Simplify any expressions as needed.
  • Isolate the variable you are solving for using algebraic operations (addition, subtraction, multiplication, division).
Using the given problem:
1. Rewrite \( \log_{1/3}(x-4)=2 \) into its exponential form \( (1/3)^2 = x-4 \).
2. Calculate \( (1/3)^2 = \frac{1}{9} \).
3. Isolate \( x \) by adding 4 to both sides: \( x = \frac{1}{9} + 4 = \frac{37}{9} \).
So, the solution for \( x \) is \( \frac{37}{9} \).

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Most popular questions from this chapter

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