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:

\(\displaystyle\Rightarrow{\frac{{{x}-{2}}}{{{6}}}}={\frac{{{1}}}{{{2}}}}\)

\(\displaystyle\Rightarrow{2}{\left({x}-{2}\right)}={6}\)

\(\displaystyle\Rightarrow{2}{x}-{4}={6}\)

\(\displaystyle\Rightarrow{2}{x}={6}+{4}\)

\(\displaystyle\Rightarrow{x}={5}\)

ANswer x=5

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:

\(\displaystyle\Rightarrow{\frac{{{x}-{2}}}{{{6}}}}={\frac{{{1}}}{{{2}}}}\)

\(\displaystyle\Rightarrow{2}{\left({x}-{2}\right)}={6}\)

\(\displaystyle\Rightarrow{2}{x}-{4}={6}\)

\(\displaystyle\Rightarrow{2}{x}={6}+{4}\)

\(\displaystyle\Rightarrow{x}={5}\)

ANswer x=5