Solve $LaTeX: \displaystyle \log_{10}(x + 17)+\log_{10}(x - 4) = 2$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{10}(x^{2} + 13 x - 68)=2$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 13 x - 68=10^{2}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 13 x - 168=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 8\right) \left(x + 21\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -21$ and $LaTeX: \displaystyle x = 8$ . The domain of the original is $LaTeX: \displaystyle \left(-17, \infty\right) \bigcap \left(4, \infty\right)=\left(4, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -21$ is not a solution. $LaTeX: \displaystyle x=8$ is a solution.