Logarithmic Equation: Solve for x

2 log₂ x - log₂ (x-1/2) = -log_1/3 3       note  that     - log_1/3 3     actually evaluates to  -[ -1 ]   = 1

So we have

2 log₂ x - log₂ (x-1/2)   = 1       and we can write

log₂ x²  - log₂ ( x - 1/2)   = 1       and by a property of logs  we have

log₂  [ ( x^2 ) / (x - 1/2)]    =  1    and in exponential form we have

2^1  =  ( x^2 ) / (x - 1/2)

2  = ( x^2) / ( x - 1/2)         multiply both sides by  x - 1/2

2( x - 1/2)  = x^2

2x - 1   = x^2       simplify

x^2 - 2x + 1  =  0        factor

(x -1 ) ( x - 1)  = 0      and setting each factor to 0   .....we have that   x = 1

Check that

2 log₂ (1) - log₂ (1-1/2)   = 1 

2 * 0   -  log₂ (1/2)  = 1

0  -  (-1)   = 1

1  =   1

Thank you so much! It was extremely helpful and explanatory for me. Thanks again.

No problem, Shades.....glad it helped.....