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

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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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.