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Chapter 10: Problem 199

In the following exercises, solve each logarithmic equation. \(\log _{3}(5 x-4)=4\)

Short Answer

Expert verified
x = 17

Step by step solution

01

Understand the given equation

The given equation is \(\( \log_{3}(5x-4)=4 \)\). This is a logarithmic equation with base 3 and argument \(\(5x - 4 \)\) set equal to 4.
02

Convert the logarithmic equation to exponential form

To solve the logarithmic equation \(\( \log_{3}(y) = 4 \)\), rewrite it in exponential form: \[\text{If } \log_{b}(y) = k, \text{then } b^k = y \]. Applying this to our equation, we get \(\( 3^4 = 5x - 4 \)\).
03

Simplify the exponential equation

Calculate \(\( 3^4 \)\): \(\( 3^4 = 81 \)\). Thus, the equation becomes: \(\( 81 = 5x - 4 \)\).
04

Solve for x

Isolate x by adding 4 to both sides: \(\( 81 + 4 = 5x \)\), which simplifies to \(\( 85 = 5x \)\). Next, divide by 5: \(\( x = \frac{85}{5} \)\), which simplifies to: \(\( x = 17 \)\).

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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 inverses of exponential functions. In simpler terms, if you know how to work with exponentials, you can easily understand logarithms. A logarithm answers the question: to what exponent must the base be raised, to produce a given number? For example, \( \log_{3}(81) = 4 \) means \( 3^{4} = 81 \). When you see a logarithm, always think: what power turns the base into the number inside the log? The general form is \( \log_{b}(y) = k \), where \b\ is the base, \y\ is the argument (the number inside), and \k\ is the logarithm (or the exponent). Logarithms have useful properties like:
  • The product rule: \( \log_{b}(mn) = \log_{b}(m) + \log_{b}(n)\)
  • The quotient rule: \( \log_{b}(m/n) = \log_{b}(m) - \log_{b}(n)\)
  • The power rule: \( \log_{b}(m^n) = n \log_{b}(m)\)
These properties help simplify complex logarithmic expressions.
Exponential Form
Converting a logarithmic equation into exponential form makes it easier to solve. Logarithmic and exponential forms are just two sides of the same coin. For the equation given in the exercise, \( \log_{3}(5x-4) = 4 \), we can convert it by using the exponential definition: \( \log_{b}(y) = k \iff b^k = y\). Therefore, \( b=3, k=4, \text{ and } y=5x-4 \) can be rewritten as \( 3^4 = 5x-4\). This method translates the logarithmic form into an exponential one that can be easily managed with basic algebra.
Solving Equations
Once the logarithmic equation is in exponential form, solving it becomes straightforward. Let's take the transformed equation \( 3^4 = 5x-4\). The first step is to calculate the power: \( 3^4 \) is 81. So, our equation becomes \( 81 = 5x-4 \). To solve for \x\, we need to isolate it:
  • Add 4 to both sides: \( 81 + 4 = 85 \)
  • The equation simplifies to \( 85 = 5x \)
  • Finally, divide by 5: \( 85/5 = x \), which simplifies to \( x = 17 \).
This step-by-step approach ensures clarity and makes it easier to check the work by substituting \( x \) back into the original equation.

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Most popular questions from this chapter

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible. \(\log _{2} 80-\log _{2} 5\)

In the following exercises, solve each equation. \(\log _{4}(7 x+15)=3\)

In the following exercises, solve each equation. \(\log _{2}(x+2)-\log _{2}(2 x+9)=-\log _{2} x\)

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places. \(2^{x}=74\)

In the following exercises, solve for \(x\). $$ \log 49=2 \log x $$
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