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

Tazmin Horton 2021-09-11 Answered
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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Expert Answer

Bentley Leach
Answered 2021-09-12 Author has 27528 answers
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
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