Question 210783
I'm assuming that the first 'x's on each side are multiplication symbols.



{{{4*3^x = 7*5^x}}} Start with the given equation.



{{{log(10,(4*3^x))=log(10,(7*5^x))}}} Take the log of both sides.



{{{log(10,(4))+log(10,(3^x))=log(10,(7))+log(10,(5^x))}}} Break up the log using the identity  {{{log(b,(A*B))=log(b,(A))+log(b,(B))}}}



{{{log(10,(3^x))=log(10,(7))+log(10,(5^x))-log(10,(4))}}} Subtract {{{log(10,(4))}}} from both sides.



{{{log(10,(3^x))-log(10,(5^x))=log(10,(7))-log(10,(4))}}} Subtract {{{log(10,(5^x))}}} from both sides.



{{{x*log(10,(3))-x*log(10,(5))=log(10,(7))-log(10,(4))}}} Rewrite the logs on the left side using the identity  {{{log(b,(x^y))=y*log(b,(x))}}}



{{{x*(log(10,(3))-log(10,(5)))=log(10,(7))-log(10,(4))}}} Factor out the GCF 'x' on the left side.



{{{x*(log(10,(3/5)))=log(10,(7/4))}}} Combine the logs using the identity {{{log(b,(A))-log(b,(B))=log(b,(A/B))}}}



{{{x=log(10,(7/4))/(log(10,(3/5)))}}} Divide both sides by {{{log(10,(3/5))}}} to isolate 'x'



{{{x=-1.0955}}} Approximate the answer (with a calculator)



So the exact answer is {{{x=log(10,(7/4))/(log(10,(3/5)))}}}, which can be simplified to *[Tex \LARGE x=\log_{\frac{3}{5}}\left(\frac{7}{4}\right)] and the approximate answer is {{{x=-1.0955}}}