Problem 123 - College Algebra

Question

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

Step-by-Step Solution

Answer

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 1: 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 \( x = 82 \).

Step 2: 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 \( x = 5 \).