Question

4^x+4^2x=7

Answer 1

\[4^{x}+4^{2x}=7\]
By theory of exponents (they add if the bases are the same),
\[4^{3x} = 7\]
Taking natural logarithm on both sides (the exponent comes down),
\[3x \ln(4) = \ln(7)\]
Moving the logarithms to one side,
\[3x = \frac{ \ln(7) }{ \ln(4) }\]
Making it explicit for x,
\[x = \frac{ \ln(7) }{ \ln(4) } * \frac{ 1 }{ 3 }\]
Simplifying and expressing the answer in decimals,
\[x = \frac{ \ln(7) }{ 3\ln(4) } \approx 0.467892\]
Both answers (the logarithmic fraction and the decimal approximation) are valid.

Answer 2

ah.ok o got the answer 0.47