Chapter 3.2, Problem 77E

To determine

To find: The value of the variable x for the given equation.

Answer to Problem 77E

The value of the variable x for the equation $\ln \left(x^2-x\right)=\ln 6$ is $x=-2$ , $x=3$.

Explanation of Solution

Given information:

The equation $\ln \left(x^2-x\right)=\ln 6$ .

Formula used:

One-to-One property states that to solve the logarithmic equation of the form $\ln x=\ln y$ where $x>0$ , $y>0$ and $x\ne 1$ , $y\ne 1$ the statement $\ln x=\ln y$ holds true if and only if $x=y$ .

Calculation:

Consider the provided equation, $\ln \left(x^2-x\right)=\ln 6$ .

Recall that One-to-One property states that to solve the logarithmic equation of the form $\ln x=\ln y$ where $x>0$ , $y>0$ and $x\ne 1$ , $y\ne 1$ the statement $\ln x=\ln y$ holds true if and only if $x=y$ .

Apply it,

  $\begin{array}{c}\ln \left(x^2-x\right)=\ln 6\\ x^2-x=6\\ x^2-x-6=0\\ x^2-3x+2x-6=0\end{array}$

Simplify it further,

  $\left(x-3\right)\left(x+2\right)=0$

Either $x=3$ or $x=-2$

Hence, the value of the variable x for the equation $\ln \left(x^2-x\right)=\ln 6$ is $x=3$ or $x=-2$ .