Precalculus Examples

$2 x \ln (x - x) = 0$

Step 1

Simplify $2 \ln (x - x)$ by moving $2$ inside the logarithm.

$x \ln ({(x - x)}^{2}) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$\ln ({(x - x)}^{2}) = 0$

Step 3

Set $x$ equal to $0$ .

$x = 0$

Step 4

Set $\ln ({(x - x)}^{2})$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $\ln ({(x - x)}^{2})$ equal to $0$ .

$\ln ({(x - x)}^{2}) = 0$

Step 4.2

Solve $\ln ({(x - x)}^{2}) = 0$ for $x$ .

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Step 4.2.1

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln ({(x - x)}^{2})} = e^{0}$

Step 4.2.2

Rewrite $\ln ({(x - x)}^{2}) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{0} = {(x - x)}^{2}$

Step 4.2.3

Solve for $x$ .

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Step 4.2.3.1

Rewrite the equation as ${(x - x)}^{2} = e^{0}$ .

${(x - x)}^{2} = e^{0}$

Step 4.2.3.2

Subtract $x$ from $x$ .

$0^{2} = e^{0}$

Step 4.2.3.3

Anything raised to $0$ is $1$ .

$0^{2} = 1$

Step 4.2.3.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$0 = \pm \sqrt{1}$

Step 4.2.3.5

Any root of $1$ is $1$ .

$0 = \pm 1$

Step 4.2.3.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.2.3.6.1

First, use the positive value of the $\pm$ to find the first solution.

$0 = 1$

Step 4.2.3.6.2

Since $0 \neq 1$ , there are no solutions.

No solution

Step 4.2.3.6.3

Next, use the negative value of the $\pm$ to find the second solution.

$0 = - 1$

Step 4.2.3.6.4

Since $0 \neq - 1$ , there are no solutions.

No solution

Step 5

The final solution is all the values that make $x \ln ({(x - x)}^{2}) = 0$ true.

$x = 0$

Step 6

Exclude the solutions that do not make $x \ln ({(x - x)}^{2}) = 0$ true.

$No solution$