Notes: `3^(x+4)=6^(2x-5)` Solve the equation.

Saturday, 12 September 2015

$\displaystyle{{3}}^{{{x}+{4}}}={{6}}^{{{2}{x}-{5}}}$

To solve, take the natural logarithm of both sides.

$\displaystyle{\ln{{\left({{3}}^{{{x}+{4}}}\right)}}}={\ln{{\left({{6}}^{{{2}{x}-{5}}}\right)}}}$

To simplify each side, apply the logarithm rule $\displaystyle{\ln{{\left({{a}}^{{m}}\right)}}}={m}\cdot{\ln{{\left({a}\right)}}}$ .

$\displaystyle{\left({x}+{4}\right)}{\ln{{\left({3}\right)}}}={\left({2}{x}-{5}\right)}{\ln{{\left({6}\right)}}}$

$\displaystyle{x}{\ln{{\left({3}\right)}}}+{4}{\ln{{\left({3}\right)}}}={2}{x}{\ln{{\left({6}\right)}}}-{5}{\ln{{\left({6}\right)}}}$

Then, bring together the terms with x on one side of the equation. Also, bring together the terms without x on the other side of the equation.

$\displaystyle{x}{\ln{{\left({3}\right)}}}-{2}{x}{\ln{{\left({6}\right)}}}=-{4}{\ln{{\left({3}\right)}}}-{5}{\ln{{\left({6}\right)}}}$

At the left...

$\displaystyle{{3}}^{{{x}+{4}}}={{6}}^{{{2}{x}-{5}}}$

To solve, take the natural logarithm of both sides.

$\displaystyle{\ln{{\left({{3}}^{{{x}+{4}}}\right)}}}={\ln{{\left({{6}}^{{{2}{x}-{5}}}\right)}}}$

To simplify each side, apply the logarithm rule $\displaystyle{\ln{{\left({{a}}^{{m}}\right)}}}={m}\cdot{\ln{{\left({a}\right)}}}$ .

$\displaystyle{\left({x}+{4}\right)}{\ln{{\left({3}\right)}}}={\left({2}{x}-{5}\right)}{\ln{{\left({6}\right)}}}$

$\displaystyle{x}{\ln{{\left({3}\right)}}}+{4}{\ln{{\left({3}\right)}}}={2}{x}{\ln{{\left({6}\right)}}}-{5}{\ln{{\left({6}\right)}}}$

Then, bring together the terms with x on one side of the equation. Also, bring together the terms without x on the other side of the equation.

$\displaystyle{x}{\ln{{\left({3}\right)}}}-{2}{x}{\ln{{\left({6}\right)}}}=-{4}{\ln{{\left({3}\right)}}}-{5}{\ln{{\left({6}\right)}}}$

At the left side, factor out the GCF.

$\displaystyle{x}{\left({\ln{{\left({3}\right)}}}-{2}{\ln{{\left({6}\right)}}}\right)}=-{4}{\ln{{\left({3}\right)}}}-{5}{\ln{{\left({6}\right)}}}$

And, isolate the x.

$\displaystyle{x}=\frac{{-{4}{\ln{{\left({3}\right)}}}-{5}{\ln{{\left({6}\right)}}}}}{{{\ln{{\left({3}\right)}}}-{2}{\ln{{\left({6}\right)}}}}}$

$\displaystyle{x}\approx{5.374}$

Therefore, the solution is $\displaystyle{x}\approx{5.374}$ .

Notes