Question 143994
{{{2^(x^3)=3^(x^2)}}} Start with the given equation



{{{log(10,(2^(x^3)))=log(10,(3^(x^2)))}}} Take the log of both sides



{{{x^3*log(10,(2))=x^2*log(10,(3))}}} Rewrite the expressions using the identity  {{{log(b,(x^y))=y*log(b,(x))}}}



{{{x^3*log(10,(2))-x^2*log(10,(3))=0}}} Subtract {{{x^2*log(10,(3))}}} from both sides



{{{x^2(x*log(10,(2))-log(10,(3)))=0}}} Factor out the GCF {{{x^2}}}



{{{x^2=0}}} or {{{x*log(10,(2))-log(10,(3))=0}}} Set each individual factor equal to zero



Let's solve the first equation {{{x^2=0}}} 



{{{x=0}}} Take the square root of both sides. So this is our first answer.


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Now let's solve the second equation {{{x*log(10,(2))-log(10,(3))=0}}} 


{{{x*log(10,(2))=log(10,(3))}}} Add {{{log(10,(3))}}}  to both sides



{{{x=log(10,(3))/log(10,(2))}}} Divide both sides by {{{log(10,(2))}}} to isolate x



{{{x=log(2,(3))}}} Use the change of base formula to simplify. So this is our other answer.




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Answer:


So the solutions are 

{{{x=0}}}  or {{{x=log(2,(3))}}}


which approximate to


{{{x=0}}}  or {{{x=1.58496}}}