Q 104. - Precalculus Enhanced with Graphing Utilities

Question

Solve the equation. 

${\log }_5\left(x^2+x+4\right)=2$

Step-by-Step Solution

Answer

The solution to the equation, ${\log }_5\left(x^2+x+4\right)=2$is, $x=\frac{-1+\sqrt{85}}{2}$.

Step 1. Given Information

The given equation to be solved is ${\log }_5\left(x^2+x+4\right)=2$.

Step 2. Formula used

Rule of change from logarithm to exponents: 

${\log }_{a}b=y\Rightarrow a^y=b$

Step 3. Finding Solution

The given equation is ${\log }_5\left(x^2+x+4\right)=2$.

Use rule of change from logarithm to exponents: 

$$\begin{gathered} {\log }_5\left(x^2+x+4\right)=2 \\ {x}^2+x+4=5^2 \\ {x}^2+x+4=25 \\ {x}^2+x-21=0 \\ x=\frac{-1\pm \sqrt{1+84}}{2} \\ x=\frac{-1\pm \sqrt{85}}{2} \end{gathered}$$

The solution to the given equation is,  $x=\frac{-1+\sqrt{85}}{2}$.