Question 85515
{{{4^(3x)=7^(x+1)}}} Start with the given equation


{{{log(10,(4^(3x)))=log(10,(7^(x+1)))}}} Take the log (which has a default base of 10) of both sides


{{{3x*log(10,4)=(x+1)log(10,7)}}} Rewrite the logarithms using the identity {{{log(b,x^a)=a*log(b,x)}}} 


{{{3x*log(10,4)-(x+1)log(10,7)=0}}} Subtract {{{(x+1)log(10,7)}}} from both sides



{{{3x*log(10,4)-(x*log(10,7)+log(10,7))=0}}} Distribute {{{log(10,7)}}}


{{{3x*log(10,4)-x*log(10,7)-log(10,7)=0}}} Distribute the negative


{{{3x*log(10,4)-x*log(10,7)=log(10,7)}}} Add {{{log(10,7)}}}


{{{x(3*log(10,4)-log(10,7))=log(10,7)}}} Factor out an x


{{{x(log(10,4^3)-log(10,7))=log(10,7)}}} Rewrite the logarithm using the identity {{{a*log(b,x)=log(b,x^a)}}}


{{{x(log(10,64)-log(10,7))=log(10,7)}}} Raise 4 to the third power


{{{x(log(10,(64/7)))=log(10,7)}}} Combine the logs using {{{log(b,x)-log(b,y)=log(b,(x/y))}}}


{{{x=log(10,7)/(log(10,(64/7)))}}} Divide both sides by {{{log(10,(64/7))}}}





{{{x=0.845098/0.961082}}} Using {{{log(10,7)=0.845098}}} and {{{log(10,(64/7))=0.961082}}} as approximations, evaluate the logs


{{{x=0.879319}}} Divide



Check:


{{{4^(3*0.879319)=7^(0.879319+1)}}} Plug in {{{x=0.879319}}}


{{{4^(2.637957)=7^(1.879319)}}}


{{{38.7443=38.7443}}} works