Question

Solve the following logarithmic equations.
ln(32x^2) − 3 ln(2) = 3

Answer 1

`x=\pm \frac{\sqrt{e^(3) } }{2}`

Explanation:

Rewrite the equation using the next propierties:

`log((1)/(x) )=-log(x)`

`ylog(x)=log(x^(y) )`

`log(x*y)=log(x)+log(y)`

`ln(32x^(2) )-ln(2^(3))=3\\ ln(32x^(2) )+ln((1)/(2^(3) ) )=3\\ln((32x^(2) )/(8))=3\\ ln(4x^(2) )=3`

Cancel logarithms by taking exp of both sides:

`e^{ln(4x^(2)) } =e^(3) \\4x^(2) =e^(3)`

Divide both sides by 4:

`x^(2) =(e^(3) )/(4)`

Take the square root of both sides:

`x=\pm \sqrt{(e^(3) )/(4) } =\pm \frac{\sqrt{e^(3) } }{2}`