# Solve the exponential equation. Express irrational solutions in exact form.

Solve the exponential equation. Express irrational solutions in exact form.
$$\displaystyle{6}^{{{1}-{4}{x}}}={5}^{{x}}$$

• Questions are typically answered in as fast as 30 minutes

### Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

un4t5o4v

Given exponential equation is
$$\displaystyle{6}^{{{1}-{4}{x}}}={5}^{{x}}$$
Taking natural log on both sides,
$$\displaystyle{\ln{{\left({6}^{{{1}-{4}{x}}}\right)}}}={\ln{{\left({5}^{{x}}\right)}}}$$
We know, $$\displaystyle{\ln{{\left({a}^{{b}}\right)}}}={b}{\ln{{\left({a}\right)}}}$$
$$\displaystyle{\left({1}-{4}{x}\right)}{\ln{{\left({6}\right)}}}={x}{\ln{{\left({5}\right)}}}$$
$$\displaystyle{\ln{{\left({6}\right)}}}-{4}{x}{\ln{{\left({6}\right)}}}={x}{\ln{{\left({5}\right)}}}$$
$$\displaystyle-{4}{x}{\ln{{\left({6}\right)}}}-{x}{\ln{{\left({5}\right)}}}=-{\ln{{\left({6}\right)}}}$$
$$\displaystyle-{x}{\left[{4}{\ln{{\left({6}\right)}}}+{\ln{{\left({5}\right)}}}\right]}=-{\ln{{\left({6}\right)}}}$$
$$\displaystyle{x}{\left[{4}{\ln{{\left({6}\right)}}}+{\ln{{\left({5}\right)}}}\right]}={\ln{{\left({6}\right)}}}$$
$$\displaystyle{x}={\frac{{{\ln{{\left({6}\right)}}}}}{{{\left[{4}{\ln{{\left({6}\right)}}}+{\ln{{\left({5}\right)}}}\right]}}}}$$
$$\displaystyle{x}\approx{0.2042}$$
Answer: The solution set is $$\{{\frac{{{\ln{{\left({6}\right)}}}}}{{{\left|{4}{\ln{{\left({6}\right)}}}+{\ln{{\left({5}\right)}}}\right|}}}}\}$$ or $$\{0.2042\}$$