Solve the exponential equation 6^{(x-3)/4}=\sqrt{6} by expressing each sid

Marvin Mccormick 2021-09-29 Answered
Solve the exponential equation \(\displaystyle{6}^{{\frac{{{x}-{3}}}{{4}}}}=\sqrt{{{6}}}\) by expressing each side as a power of the same base and then equating exponents.

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

doplovif
Answered 2021-09-30 Author has 18799 answers
Given: \(\displaystyle{6}^{{{\frac{{{x}-{3}}}{{{4}}}}}}=\sqrt{{{6}}}\)
for solving this equation we make base same both side then equate power
we know that
\(\displaystyle{6}={\left(\sqrt{{{6}}}\right)}^{{2}}\)
so,
in place of 6 put \(\displaystyle{\left(\sqrt{{{6}}}\right)}^{{2}}\)
\(\displaystyle{\left({\left(\sqrt{{{6}}}\right)}^{{2}}\right)}^{{{\frac{{{x}-{3}}}{{{4}}}}}}=\sqrt{{{6}}}\)
\(\displaystyle{\left(\sqrt{{{6}}}\right)}^{{{\left({2}\times{\frac{{{x}-{3}}}{{{4}}}}\right)}}}=\sqrt{{{6}}}\)
\(\displaystyle{\left(\sqrt{{{6}}}\right)}^{{{\frac{{{x}-{3}}}{{{2}}}}}}={\left(\sqrt{{{6}}}\right)}^{{1}}\)
now here, base is same so power also be same
\(\displaystyle{\frac{{{x}-{3}}}{{{2}}}}={1}\)
\(\displaystyle{x}-{3}={2}\)
\(\displaystyle{x}={3}+{2}\)
\(\displaystyle{x}={5}\)
hence, solution of given expression is 5.
Have a similar question?
Ask An Expert
49
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more
...