Question

solve each equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places:

${\log }_2(3x+2)-{\log }_4\left(x\right)=3$

Short answer

The value of x satisfying the equation is$x=0.078andx=5.69$

Step-by-step solution

Given information

We are given a equation${\log }_2(3x+2)-{\log }_4\left(x\right)=3$

Change the base

We have,

$$\begin{gathered} {\log }_2(3x+2)-{\log }_4\left(x\right)=3 \\ {\log }_2(3x+2)-\frac{{\log }_2\left(x\right)}{{\log }_2{\left(2\right)}^2}=3 \\ {\log }_2(3x+2)-\frac{{\log }_2\left(x\right)}{2}=3 \\ 2{\log }_2(3x+2)-{\log }_2\left(x\right)=6 \end{gathered}$$

Solve the equation

We get,

$$\begin{gathered} 2{\log }_2(3x+2)-{\log }_2\left(x\right)=6 \\ l{\mathrm{og}}_2{(3x+2)}^2-{\log }_2\left(x\right)=6 \\ {\log }_2\frac{{(3x+2)}^2}{x}=6 \end{gathered}$$

Now use the formula $y={\log }_a\left(x\right)\iff x=a^y$

$$\begin{gathered} \frac{{(3x+2)}^2}{x}=2^6 \\ {(3x+2)}^2=64x9x^2+12x+4=64x \\ 9x^2-52x+4=0 \end{gathered}$$

Find the value of x

We get,

$$\begin{gathered} 9x^2-52x+4=0 \\ x=\frac{52\pm \sqrt{{52}^2-4\left(9\right)\left(4\right)}}{18} \\ x=\frac{52\pm \sqrt{2704-144}}{18} \\ x=\frac{52\pm 50.59}{18} \\ x=\frac{52+50.59}{18}orx=\frac{52-50.59}{18} \\ x=5.69orx=0.078 \end{gathered}$$