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

Solve the exponential equation by expressing each side as a power of the same base and then equating exponents.
$$\displaystyle{7}^{{{\frac{{{x}-{2}}}{{{6}}}}}}=\sqrt{{{7}}}$$

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Bentley Leach
To solve the exponential equation: $$\displaystyle{7}^{{{\frac{{{x}-{2}}}{{{6}}}}}}=\sqrt{{{7}}}$$
Solution:
$$\displaystyle{7}^{{{\frac{{{x}-{2}}}{{{6}}}}}}=\sqrt{{{7}}}$$
On simplifying further we get:
$$\displaystyle{7}^{{{\frac{{{x}-{2}}}{{{6}}}}}}=\sqrt{{{7}}}$$
$$\displaystyle\Rightarrow{7}^{{{\frac{{{x}-{2}}}{{{6}}}}}}={7}^{{{\frac{{{1}}}{{{2}}}}}}$$
Now, base(7) is same both sides, so equating exponents:
$$\displaystyle\Rightarrow{\frac{{{x}-{2}}}{{{6}}}}={\frac{{{1}}}{{{2}}}}$$
$$\displaystyle\Rightarrow{2}{\left({x}-{2}\right)}={6}$$
$$\displaystyle\Rightarrow{2}{x}-{4}={6}$$
$$\displaystyle\Rightarrow{2}{x}={6}+{4}$$
$$\displaystyle\Rightarrow{x}={5}$$