Chapter 8.6, Problem 1E

To determine

To find: The solution of equation.

Answer to Problem 1E

The solution of the equation $lo{g}_2\left(3x+2\right)=lo{g}_3\left(12x+3\right)$ is $1$ .

Explanation of Solution

Given information:

The expression is

  $lo{g}_2\left(3x+2\right)=lo{g}_3\left(12x+3\right)$

Calculation:

Consider following equation

  $lo{g}_2\left(3x+2\right)=lo{g}_3\left(12x+3\right)$

Rewrite the equation by using the formula as follows:

  $lo{g}_a\left(b\right)=\frac{\log \left(b\right)}{\log (a)}$

So,

  $\begin{array}{l}lo{g}_2\left(3x+2\right)=lo{g}_3\left(12x+3\right)\\ \frac{log\left(3x+2\right)}{log\left(2\right)}=\frac{log\left(12x+\text{ 3}\right)}{log\left(3\right)}\end{array}$

Use TI-83 calculator to solve equation

Press $Y=$

1. and enter the equations, the screenshot is given as below:

  

2. Press window button, and set window as given below:

  

3. Press graph button, the screenshot is given as below

  

To find approximate ordered pair of the point at which curves intersect, uses intersect features Press $2^{nd}$ and CALC button. The screenshot is given as below:

  

Press 5 then press ENTER to set first curve and press ENTER second curve values. Again press ENTER twice to get interception point. Window that shows intersectionpoint is displayed below:

  

Hence, solution of the equation $lo{g}_2\left(3x+2\right)=lo{g}_3\left(12x+3\right)$ is $1$ .

Now check the solution by using table feature, for this Press 2nd and TABLE button. The screenshot is given as below

  

According to above table at $x=0$ both functions have y-value of $1$ ,

Therefore, solution of the equation is $1$ .