Question

What is the solution to log^2 (2x^3 -8)-2log ^2 x= log^2x

Answer 1

Hey

$log_{2}(2 {x}^{3} - 8) - 2 log_{2}(x) = log_{2}(x)$

Transposing log x to other side :

$= > log_{2}(2 {x}^{3} - 8) = log_{2}(x) + 2 log_{2}(x)$
Using Logarithmic Property :

$= > log_{2}(2 {x}^{3} - 8) = log_{2}( {x}^{3} )$

Raising to the power 2 :

${x}^{3} - 8 = 0$

$= > (x - 2)( {x}^{2} + 2x + 4) = 0$

$= > x = 2$

Hence, [ x = 2 ] is the only real solution !