Question

solve for x.
\log_{2}(x - 2) = -\log_{2}(x - 5) + 2
if there is more than one solution, separate them with commas.
if there is no solution, click on
o solution\.

Explanation:

Step1: Move log term to left

Use logarithm property \( - \log_b a=\log_b \frac{1}{a} \), rewrite equation:
\( \log_2 (x - 2)+\log_2 (x - 5)=2 \)
Apply \( \log_b m+\log_b n=\log_b (mn) \):
\( \log_2 [(x - 2)(x - 5)] = 2 \)

Step2: Convert to exponential form

Recall \( \log_b y = k \implies y = b^k \). Here, \( b = 2 \), \( k = 2 \), so:
\( (x - 2)(x - 5)=2^2 = 4 \)

Step3: Expand and solve quadratic

Expand left side: \( x^2 - 7x + 10 = 4 \)
Simplify: \( x^2 - 7x + 6 = 0 \)
Factor: \( (x - 1)(x - 6)=0 \)
Solutions: \( x = 1 \) or \( x = 6 \)

Step4: Check domain

Logarithm requires \( x - 2>0 \) and \( x - 5>0 \implies x>5 \).

• \( x = 1 \): \( 1 - 5=-4<0 \) (invalid).
• \( x = 6 \): \( 6 - 2 = 4>0 \), \( 6 - 5 = 1>0 \) (valid).

Answer:

6