Solving Logarithmic equation

Question

(1/(log₂(x+2)²) + (1/(log₂(x-2)²) = 5 /12

Answer 1

Logarithms reduce the complexity of the expression...

log(25) = log(5²) = 2log(5)

[ 1/log₂(x+2)² ] = 1/(2log₂(x+2))

[ 1/log₂(x-2)² ] = 1/(2log₂(x-2))

Multiplying by 2...

1/(log₂(x+2)) + 1/(log₂(x-2)) = 5/6

We know that x - 2 > 0...

Since the right hand side is a rational number --> particular quantities of x+2 or x-2 must be simple solutions for the logarithm base 2.

x = 3 --> log₂(x+2) is not simple

x = 4 --> log₂(x+2) is not simple

x = 5 --> log₂(x+2) is not simple

x = 6 --> log₂(x+2) is simple and yields 3

log₂(x-2) is simple and yields 2

Does 1/3 + 1/2 equal 5/6. Yes!