6^x+2=4^x

6^x+2=4^x

Question
Logarithms
asked 2020-11-14
\(\displaystyle{6}^{{x}}+{2}={4}^{{x}}\)

Answers (1)

2020-11-15

Take the natural logarithm of both sides:
\(\displaystyle{{\ln{{6}}}^{{x}}+}{2}={{\ln{{4}}}^{{x}}}\)
Apply Power Property of logarithms: \(\displaystyle{\log{{b}}}{x}^{{a}}={a}{\log{{b}}}{x}\)
\(\displaystyle{\left({x}+{2}\right)}{\ln{{6}}}={x}{\ln{{4}}}\)
\(\displaystyle{x}{\ln{{6}}}+{2}{\ln{{6}}}={x}{\ln{{4}}}\)
Subtract \(x\ln6\) from both sides:
\(\displaystyle{2}{\ln{{6}}}={x}{\ln{{4}}}−{x}{\ln{{6}}}\)
Factor out x:
\(\displaystyle{2}{\ln{{6}}}={\left({\ln{{4}}}−{\ln{{6}}}\right)}{x}\)
Divide both sides by \(\displaystyle{\ln{{4}}}−{\ln{{6}}}{\ln{{4}}}−{\ln{{6}}}:\)
\(\displaystyle{2}\frac{{\ln{{6}}}}{{{\ln{{4}}}−{\ln{{6}}}}}={x}\)
or
\(\displaystyle{x}={2}\frac{{\ln{{6}}}}{{{\ln{{4}}}−{\ln{{6}}}}}≈−{8.84}\)

0

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