Question 244839
{{{2^(4x)*3^x=100}}} Start with the given equation.



{{{log(10,(2^(4x)*3^x))=log(10,(100))}}} Take the log of both sides.



{{{log(10,(2^(4x)*3^x))=2}}} Evaluate the log base 10 of 100 to get 2.



{{{log(10,(2^(4x)))+log(10,(3^x))=2}}} Break up the log using the identity  {{{log(b,(A*B))=log(b,(A))+log(b,(B))}}}



{{{4x*log(10,(2))+x*log(10,(3))=2}}} Pull down the exponents using the identity  {{{log(b,(x^y))=y*log(b,(x))}}}



{{{x(4*log(10,(2))+log(10,(3)))=2}}} Factor out the GCF 'x'



{{{x=2/(4*log(10,(2))+log(10,(3)))}}} Divide both sides by {{{4*log(10,(2))+log(10,(3))}}}



{{{x=2/(log(10,(2^4))+log(10,(3)))}}} Place the coefficient as the exponent using the identity  {{{y*log(b,(x))=log(b,(x^y))}}}



{{{x=2/(log(10,(16))+log(10,(3)))}}} Raise 2 to the 4th power to get 16



{{{x=2/(log(10,(16*3)))}}} Combine the logs using the identity {{{log(b,(A))+log(b,(B))=log(b,(A*B))}}}



{{{x=2/log(10,(48))}}} Multiply



So the solution is {{{x=2/log(10,(48))}}} which approximates to {{{x=1.18959727}}}