Question 149457
{{{f(x)= 6^(x-1) }}} Start with the given function.



{{{x= 6^(f(x)-1) }}} Switch x and f(x).



{{{log(10,(x))= log(10,(6^(f(x)-1)))}}} Take the log of both sides.



{{{log(10,(x))= (f(x)-1)*log(10,(6))}}} Rewrite the right side using the identity  {{{log(b,(x^y))=y*log(b,(x))}}}



{{{log(10,(x))= f(x)*log(10,(6))-log(10,(6))}}} Distribute



{{{log(10,(x))+log(10,(6))= f(x)*log(10,(6))}}} Add {{{log(10,(6))}}} to both sides.



{{{log(10,(6x))= f(x)*log(10,(6))}}} Combine the logs using the identity {{{log(b,(A))+log(b,(B))=log(b,(A*B))}}}



{{{log(10,(6x))/log(10,(6))= f(x)}}} Divide both sides by {{{log(10,(6))}}} to isolate f(x)



{{{log(6,(6x))= f(x)}}} Use the change of base formula to rewrite the left side.



So the inverse function is *[Tex \LARGE f^{-1}\left(x\right)=\log_{6}\left(6x\right)]



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{{{(1)/(2^x) = 5}}} Start with the given equation.



{{{1= 5(2^x)}}} Multiply both sides by {{{2^x}}}.



{{{1/5=2^x}}} Divide both sides by {{{5}}}.



{{{log(10,(1/5))=log(10,(2^x))}}}  Take the log of both sides.



{{{log(10,(1/5))=x*log(10,(2))}}}  Rewrite the right side using the identity  {{{log(b,(x^y))=y*log(b,(x))}}}



{{{log(10,(1/5))/log(10,(2))=x}}}  Divide both sides by {{{log(10,(2))}}} to isolate x



{{{log(2,(1/5))=x}}} Use the change of base formula to rewrite the left side.



So the answer is {{{x=log(2,(1/5))}}} which approximates to {{{x=-2.32192}}}