Question

which logarithmic equation has the same solution as $x - 4 = 2^3$?
$\circ$ $\log 3^2 = (x - 4)$
$\circ$ $\log 2^3 = (x - 4)$
$\circ$ $\log_2(x - 4) = 3$
$\circ$ $\log_3(x - 4) = 2$

Explanation:

Step1: Solve the original equation

First, solve \( x - 4 = 2^3 \). Calculate \( 2^3 = 8 \), so \( x - 4 = 8 \), then \( x = 12 \).

Step2: Analyze each option

• Option 1: \( \log 3^2=(x - 4) \). Calculate \( \log 9\approx1.954 \), and \( x-4 = 8 \) (from original solution), not equal.
• Option 2: \( \log 2^3=(x - 4) \). \( \log 8\approx0.903 \), and \( x - 4 = 8 \), not equal.
• Option 3: \( \log_2(x - 4)=3 \). Convert to exponential form: \( x - 4 = 2^3 \), which is the same as the original equation's left - hand side relation. Substitute \( x = 12 \), \( \log_2(12 - 4)=\log_28 = 3 \), which holds.
• Option 4: \( \log_3(x - 4)=2 \). Convert to exponential form: \( x - 4 = 3^2=9 \), then \( x = 13 \), which is different from the solution of the original equation (\( x = 12 \)).

Answer:

\( \log_2(x - 4)=3 \) (the third option: \( \boldsymbol{\log_2(x - 4)=3} \))