# Exercise ID334

Algebra → Logarithm → Logarithm equations
[Level: ] [Number of helps: 0] [Number of pictures: 0] [Number of steps: 13] [Number of characters: 242]

Solve the following equation:

${\mathrm{log}}_{10}\left({x}^{2}+19\right)-{\mathrm{log}}_{10}\left(x-8\right)=2$

${\mathrm{log}}_{10}\left({x}^{2}+19\right)-{\mathrm{log}}_{10}\left(x-8\right)={2}\phantom{\rule{0ex}{0ex}}$ $\phantom{\rule{0ex}{0ex}}$${\mathrm{log}}_{10}\left({x}^{2}+19\right)-{\mathrm{log}}_{10}\left(x-8\right)={l}{o}{{g}}_{10}{{10}}^{2}\phantom{\rule{0ex}{0ex}}$ ${l}{o}{{g}}_{10}\left({x}^{2}+19\right)-{l}{o}{{g}}_{10}\left(x-8\right)={\mathrm{log}}_{10}\mathrm{100}\phantom{\rule{0ex}{0ex}}$ ${l}{o}{{g}}_{10}\frac{{x}^{2}+19}{x-8}={\mathrm{log}}_{10}100\phantom{\rule{0ex}{0ex}}$ $\frac{{x}^{2}+19}{x-8}=100\phantom{\rule{0ex}{0ex}}$ ${x}^{2}+19=100\left(x-8\right)\phantom{\rule{0ex}{0ex}}$ ${x}^{2}+19=100x-800\phantom{\rule{0ex}{0ex}}$ ${x}^{2}+19-100x+800=0\phantom{\rule{0ex}{0ex}}$ ${x}^{2}-100x+819=0\phantom{\rule{0ex}{0ex}}$ ${x}_{1,2}=\frac{100-\sqrt{10000-3276}}{2}\phantom{\rule{0ex}{0ex}}$ $\phantom{\rule{0ex}{0ex}}$${x}_{1,2}=\frac{100±\sqrt{6724}}{2}\phantom{\rule{0ex}{0ex}}$ ${x}_{1,2}=\frac{100±82}{2}\phantom{\rule{0ex}{0ex}}$

$lo{g}_{a}a=1$

$lo{g}_{a}{a}^{n}=n$

$lo{g}_{a}\left(\frac{x}{y}\right)=lo{g}_{a}x-lo{g}_{a}y$