Notes: `1/3log_5(12x)=2` Solve the equation. Check for extraneous solutions.

Wednesday, 12 October 2016

$\displaystyle\frac{{1}}{{3}}{\log}_{{5}}{\left({12}{x}\right)}={2}$ Solve the equation. Check for extraneous solutions.

To evaluate the given equation $\displaystyle\frac{{1}}{{3}}{\log}_{{5}}{\left({12}{x}\right)}={2}$ , we may apply logarithm property: $\displaystyle{n}\cdot{\log}_{{b}}{\left({x}\right)}={\log}_{{b}}{\left({{x}}^{{n}}\right)}$ .

$\displaystyle{\log}_{{5}}{\left({{\left({12}{x}\right)}}^{{\frac{{1}}{{3}}}}\right)}={2}$

Take the "log" on both sides to be able to apply the logarithm property: $\displaystyle{{a}}^{{{\log}_{{a}}{\left({x}\right)}}}={x}$ .

$\displaystyle{{5}}^{{{\log}_{{5}}{\left({{\left({12}{x}\right)}}^{{\frac{{1}}{{3}}}}\right)}}}={{5}}^{{{2}}}$

$\displaystyle{{\left({12}{x}\right)}}^{{\frac{{1}}{{3}}}}={25}$

Cubed both sides to cancel out the fractional exponent.

$\displaystyle{{\left({{\left({12}{x}\right)}}^{{\frac{{1}}{{3}}}}\right)}}^{{3}}={{\left({25}\right)}}^{{3}}$

$\displaystyle{{\left({12}{x}\right)}}^{{\frac{{1}}{{3}}\cdot{3}}}={15625}$

$\displaystyle{{\left({12}{x}\right)}}^{{\frac{{3}}{{3}}}}={15625}$

$\displaystyle{12}{x}={15625}$

Divide both sides by $\displaystyle{12}$ .

$\displaystyle\frac{{{12}{x}}}{{12}}=\frac{{{15625}}}{{12}}$

$\displaystyle{x}=\frac{{{15625}}}{{12}}$

Checking: Plug-in $\displaystyle{x}=\frac{{{15625}}}{{12}}$ on $\displaystyle\frac{{1}}{{3}}{\log}_{{5}}{\left({12}{x}\right)}={2}$

$\displaystyle\frac{{1}}{{3}}{\log}_{{5}}{\left({12}\cdot\frac{{{15625}}}{{12}}\right)}=?{2}$

$\displaystyle\frac{{1}}{{3}}{\log}_{{5}}{\left({15625}\right)}=?{2}$

$\displaystyle{\log}_{{5}}{\left({{15625}}^{{\frac{{1}}{{3}}}}\right)}=?{2}$

$\displaystyle{\log}_{{5}}{\left({\sqrt[{{3}}]{{{15625}}}}\right)}=?{2}$

$\displaystyle{\log}_{{5}}{\left({25}\right)}=?{2}$

$\displaystyle{\log}_{{5}}{\left({{5}}^{{2}}\right)}=?{2}$

$\displaystyle{2}{\log}_{{5}}{\left({5}\right)}=?{2}$

$\displaystyle{2}\cdot{1}=?{2}$

$\displaystyle{2}={2}\ldots$