Solve for the following exponential equations. Use the natural logarithm in your

Dottie Parra 2021-09-23 Answered
Solve for the following exponential equations. Use the natural logarithm in your answer(where applicable) for full credit. Use rules for exponents, factor and simplify.
\(\displaystyle{4}^{{{1}-{x}}}={3}^{{{2}{x}+{5}}}\)

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Expert Answer

Aamina Herring
Answered 2021-09-24 Author has 4671 answers
We have,
\(\displaystyle{4}^{{{1}-{x}}}={3}^{{{2}{x}+{5}}}\)
Taking logarithm on both of the side and solving further, we get result as
\(\displaystyle{\left({1}-{x}\right)}{\log{{4}}}={\left({2}{x}+{5}\right)}{\log{{3}}}\)
\(\displaystyle{\log{{4}}}-{\left({\log{{4}}}\right)}{x}={\left({2}{\log{{3}}}\right)}{x}+{5}{\log{{3}}}\)
Taking the term related x on one of the side,
\(\displaystyle{\left({\log{{4}}}-{5}{\log{{3}}}\right)}={\left({2}{\log{{3}}}+{\log{{4}}}\right)}{x}\)
\(\displaystyle{\log{{4}}}-{{\log{{3}}}^{{5}}=}{\left({{\log{{3}}}^{{2}}+}{\log{{4}}}\right)}{x}\)
\(\displaystyle{x}={\frac{{{\left({\log{{4}}}-{\log{{243}}}\right)}}}{{{\left({\log{{9}}}+{\log{{4}}}\right)}}}}\)
Using the logarithm property \(\displaystyle{\log{{m}}}-{\log{{n}}}={\log{{\left({\frac{{{m}}}{{{n}}}}\right)}}}\) and \(\displaystyle{\log{{m}}}+{\log{{n}}}={\log{{\left({m}\cdot{n}\right)}}}\) and solving further, we get the result as
\(\displaystyle{x}={\frac{{{\left({\log{{4}}}-{\log{{243}}}\right)}}}{{{\left({\log{{9}}}+{\log{{4}}}\right)}}}}\)
\(\displaystyle{x}={\frac{{{\log{{\left({\frac{{{4}}}{{{243}}}}\right)}}}}}{{{\log{{36}}}}}}\)
\(\displaystyle{x}={{\log}_{{{36}}}{\left({\frac{{{4}}}{{{243}}}}\right)}}\)
Hence, value of x will be \(\displaystyle{x}={{\log}_{{{36}}}{4243}}\),
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