Answer:
[tex]x=\pm \frac{\sqrt{e^{3} } }{2}[/tex]
Step-by-step explanation:
Rewrite the equation using the next propierties:
[tex]log(\frac{1}{x} )=-log(x)[/tex]
[tex]ylog(x)=log(x^{y} )[/tex]
[tex]log(x*y)=log(x)+log(y)[/tex]
[tex]ln(32x^{2} )-ln(2^{3})=3\\ ln(32x^{2} )+ln(\frac{1}{2^{3} } )=3\\ln(\frac{32x^{2} }{8})=3\\ ln(4x^{2} )=3[/tex]
Cancel logarithms by taking exp of both sides:
[tex]e^{ln(4x^{2}) } =e^{3} \\4x^{2} =e^{3}[/tex]
Divide both sides by 4:
[tex]x^{2} =\frac{e^{3} }{4}[/tex]
Take the square root of both sides:
[tex]x=\pm \sqrt{\frac{e^{3} }{4} } =\pm \frac{\sqrt{e^{3} } }{2}[/tex]
