Move all the logarithms on the left hand side, and all the constants on the other:
[tex] \log_2(x-2) - \log_2(x) = -1 [/tex]
Use the rule of logarithms
[tex] \log_a(b) - \log_a(c) = \log_a\left(\dfrac{b}{c}\right) [/tex]
To rewrite the equation as
[tex] \log_2\left(\dfrac{x-2}{x}\right) = -1 [/tex]
Evaluate 2 to the power of each side:
[tex] \dfrac{x-2}{x} = 2^{-1} = \dfrac{1}{2} [/tex]
Multiply both sides by 2x:
[tex] 2(x-2) = x \iff 2x-4 = x \iff x = 4 [/tex]
1 + log₂(x - 2) = log₂(x)
1 = log₂(x) - log₂(x - 2)
1 = log₂[tex]\frac{x}{x - 2}[/tex]
2¹ = [tex]\frac{x}{x - 2}[/tex]
2(x - 2) = x
2x - 4 = x
-4 = -x
4 = x
Answer: 4
