1 + log_(2)(x - 2) = log_(2)x

Question

`1 + log_(2)(x - 2) = log_(2)x`

Answer 1

Move all the logarithms on the left hand side, and all the constants on the other:

`\log_2(x-2) - \log_2(x) = -1`

Use the rule of logarithms

`\log_a(b) - \log_a(c) = \log_a\left((b)/(c)\right)`

To rewrite the equation as

`\log_2\left((x-2)/(x)\right) = -1`

Evaluate 2 to the power of each side:

`(x-2)/(x) = 2^(-1) = (1)/(2)`

Multiply both sides by 2x:

`2(x-2) = x \iff 2x-4 = x \iff x = 4`

Answer 2

1 + log₂(x - 2) = log₂(x)

1 = log₂(x) - log₂(x - 2)

1 = log₂
`(x)/(x - 2)`

2¹ =
`(x)/(x - 2)`

2(x - 2) = x

2x - 4 = x

-4 = -x

4 = x

Answer: 4