Solve the exponential equation by expressing each side as a power of the same ba

Isa Trevino 2021-09-22 Answered
Solve the exponential equation by expressing each side as a power of the same base and then equating exponents.
\(\displaystyle{3}^{{{1}-{x}}}={\frac{{{1}}}{{{27}}}}\)

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Expert Answer

ensojadasH
Answered 2021-09-23 Author has 6526 answers
Our Aim is to solve the exponential equation by express sing each side as a power of the same base and then equating exponents:
\(\displaystyle{3}^{{{1}-{x}}}={\frac{{{1}}}{{{27}}}}-{\left({i}\right)}\)
Since, from equation - (i), we have:
\(\displaystyle{3}^{{{1}-{x}}}={\frac{{{1}}}{{{27}}}}-{\left({i}\right)}\)
\(\displaystyle\Rightarrow{3}^{{-{x}+{1}}}={\frac{{{1}}}{{{27}}}}\)
Solving exponent by taking logartihm of the above equation:
\(\displaystyle\Rightarrow{\log{{\left[{3}^{{-{x}+{1}}}\right]}}}={\log{{\left[{\frac{{{1}}}{{{27}}}}\right]}}}\)
\(\displaystyle\Rightarrow{\left(-{x}+{1}\right)}\times{\log{{\left({3}\right)}}}={\log{{\left[{\frac{{{1}}}{{{27}}}}\right]}}}\)
\(\displaystyle\Rightarrow{\left(-{x}+{1}\right)}={\frac{{{\log{{\left[{\frac{{{1}}}{{{27}}}}\right]}}}}}{{{\log{{\left({3}\right)}}}}}}\)
\(\displaystyle\Rightarrow{\left(-{x}+{1}\right)}=-{3}\)
\(\displaystyle\Rightarrow-{x}=-{3}-{1}\)
\(\displaystyle\Rightarrow-{x}=-{4}\)
Anwer -x=-4
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