# 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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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