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Chapter 4: Problem 123

Explain the differences between solving \(\log _{3}(x-1)=4\) and \(\log _{3}(x-1)=\log _{3} 4\)

Short Answer

Expert verified
For \( \log_{3}(x-1) = 4 \), x equals 82. When \( \log_{3}(x-1) = \log_{3} 4 \), x equals 5. This demonstrates that different logarithmic expressions yield different solutions.

Step by step solution

01

Solving \(\log_{3}(x-1) = 4\)

In this equation, use the property of logarithms to rewrite the equation. This implies that \( 3^{4} = x - 1 \). That is, \( 81 = x - 1 \). Solving for x, we get \( x = 82 \).
02

Solving \( \log_{3}(x-1) = \log_{3} 4 \)

In this equation, since both logarithms have the same base, it concludes that \( x - 1 = 4 \). Solving for x we get \( x = 5 \).

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Most popular questions from this chapter

Evaluate or simplify each expression without using a calculator. $$ \ln \frac{1}{e^{7}} $$

Exercises \(134-136\) will help you prepare for the material covered in the next section. Solve: \(x(x-7)=3\)

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