Question 111726
{{{log(10,(16^x)) + log(10,(4^x)) + log(10,(2^x)) = 7}}} Start with the given expression




{{{log(10,((2^4)^x)) + log(10,((2^2)^x)) + log(10,(2^x)) = 7}}} Rewrite 16 as {{{2^4}}} and 4 as {{{2^2}}}


{{{log(10,(2^(4x))) + log(10,(2^(2x))) + log(10,(2^x)) = 7}}} Distribute and multiply the exponents


{{{log(10,(2^(4x))(2^(2x))(2^x)) = 7}}} Combine the logs using the identity {{{log(b,(x))+log(b,(y))=log(b,(x*y))}}}





{{{log(10,(2^(4x+2x+x))) = 7}}} Combine the expression using {{{(b^x)(b^y)=b^(x+y)}}}



{{{log(10,(2^(7x))) = 7}}} Add the exponents



{{{7x*log(10,(2)) = 7}}} Rewrite the left side using the identity {{{log(b,(x^y))=y*log(b,(x))}}}



{{{x= 7/(7*log(10,(2)))}}} Divide both sides by {{{7*log(10,(2))}}}



{{{x= 1/(log(10,(2)))}}} Reduce