Question 149398
I'll do the first one to get you started.


a)


{{{6^(x-3) = 2^x}}} Start with the given equation.



{{{log(10,(6^(x-3))) = log(10,(2^x))}}} Take the log of both sides.



{{{(x-3)log(10,(6)) = x*log(10,(2))}}} Rewrite  both sides using the identity  {{{log(b,(x^y))=y*log(b,(x))}}}



{{{x*log(10,(6))-3log(10,(6)) = x*log(10,(2))}}} Distribute.



{{{x*log(10,(6))-3log(10,(6))-x*log(10,(2))=0}}} Subtract {{{x*log(10,(2))}}} from both sides.



{{{x*log(10,(6))-x*log(10,(2))=3log(10,(6))}}} Add {{{3log(10,(6))}}} to both sides.




{{{x(log(10,(6))-log(10,(2)))=3log(10,(6))}}} Factor out the GCF "x"



{{{x(log(10,(6/2)))=3log(10,(6))}}} Combine the logs using the identity {{{log(b,(A))-log(b,(B))=log(b,(A/B))}}}



{{{x(log(10,(3)))=3log(10,(6))}}} Divide.



{{{x=3log(10,(6))/(log(10,(3)))}}} Divide both sides by {{{log(10,(3))}}} to isolate x



{{{x=log(10,(6^3))/(log(10,(3)))}}} Rewrite the expression using the identity  {{{y*log(b,(x))=log(b,(x^y))}}}




{{{x=log(10,(216))/(log(10,(3)))}}} Raise 6 to the 3rd power to get 216



{{{x=log(3,(216))}}} Use the change of base formula to rewrite the right side.



So the exact answer is {{{x=log(3,(216))}}} which approximates to {{{x=4.892789}}}