Solve for: log_{5}(4x+3)=log_{5}(9)

Expression: $|{x}^{2}-8x+14|=2$

Use the absolute value definition to rewrite the absolute value equation as two separate equations

$\begin{array} { l }{x}^{2}-8x+14=2,\\{x}^{2}-8x+14=-2\end{array}$

Solve the equation for $x$

$\begin{array} { l }x=2,\\x=6,\\{x}^{2}-8x+14=-2\end{array}$

Solve the equation for $x$

$\begin{array} { l }x=2,\\x=6,\\x=4\end{array}$

The equation has $3$ solutions

$\begin{array} { l }x_1=2,& x_2=4,& x_3=6\end{array}$