Solve the exponential equation 32^x=8 by expressing each side as a power o

abondantQ 2021-10-08 Answered
Solve the exponential equation \(\displaystyle{32}^{{x}}={8}\) by expressing each side as a power of the same base and then equating exponents.

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Expert Answer

un4t5o4v
Answered 2021-10-09 Author has 15331 answers
To determine:
To solve \(\displaystyle{32}^{{x}}={8}\) by expressing each side as a power of the same base and then equating exponents.
Calculation:
We know, \(\displaystyle{32}={2}\times{2}\times{2}\times{2}\times{2}\)
\(\displaystyle\Rightarrow{32}={2}^{{5}}\)
Also, \(\displaystyle{8}={2}\times{2}\times{2}\)
\(\displaystyle\Rightarrow{8}={2}^{{3}}\)
Plugging these values in the given equation, we get,
\(\displaystyle{\left({2}^{{5}}\right)}^{{x}}={2}^{{3}}\)
\(\displaystyle\Rightarrow{2}^{{{5}{x}}}={2}^{{3}}\)
If base are same the equating powers, we get,
\(\displaystyle{5}{x}={3}\)
\(\displaystyle\Rightarrow{x}={\frac{{{3}}}{{{5}}}}\)
Hence, \(\displaystyle{x}={\frac{{{3}}}{{{5}}}}\) is the solution of given equation.
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