Finite Math Examples

$\frac{\log_{3} ({(1 - x)}^{2})}{x^{2} - 3} = 0$

Step 1

Set the numerator equal to zero.

$\log_{3} ({(1 - x)}^{2}) = 0$

Step 2

Solve the equation for $x$ .

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Step 2.1

Write in exponential form.

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Step 2.1.1

For logarithmic equations, $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ such that $x > 0$ , $b > 0$ , and $b \neq 1$ . In this case, $b = 3$ , $x = {(1 - x)}^{2}$ , and $y = 0$ .

$b = 3$

$x = {(1 - x)}^{2}$

$y = 0$

Step 2.1.2

Substitute the values of $b$ , $x$ , and $y$ into the equation $b^{y} = x$ .

$3^{0} = {(1 - x)}^{2}$

Step 2.2

Solve for $x$ .

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Step 2.2.1

Rewrite the equation as ${(1 - x)}^{2} = 3^{0}$ .

${(1 - x)}^{2} = 3^{0}$

Step 2.2.2

Anything raised to $0$ is $1$ .

${(1 - x)}^{2} = 1$

Step 2.2.3

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

$1 - x = \pm \sqrt{1}$

Step 2.2.4

Any root of $1$ is $1$ .

$1 - x = \pm 1$

Step 2.2.5

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

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Step 2.2.5.1

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

$1 - x = 1$

Step 2.2.5.2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.2.5.2.1

Subtract $1$ from both sides of the equation.

$- x = 1 - 1$

Step 2.2.5.2.2

Subtract $1$ from $1$ .

$- x = 0$

Step 2.2.5.3

Divide each term in $- x = 0$ by $- 1$ and simplify.

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Step 2.2.5.3.1

Divide each term in $- x = 0$ by $- 1$ .

$\frac{- x}{- 1} = \frac{0}{- 1}$

Step 2.2.5.3.2

Simplify the left side.

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Step 2.2.5.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{0}{- 1}$

Step 2.2.5.3.2.2

Divide $x$ by $1$ .

$x = \frac{0}{- 1}$

Step 2.2.5.3.3

Simplify the right side.

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Step 2.2.5.3.3.1

Divide $0$ by $- 1$ .

$x = 0$

Step 2.2.5.4

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

$1 - x = - 1$

Step 2.2.5.5

Move all terms not containing $x$ to the right side of the equation.

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Step 2.2.5.5.1

Subtract $1$ from both sides of the equation.

$- x = - 1 - 1$

Step 2.2.5.5.2

Subtract $1$ from $- 1$ .

$- x = - 2$

Step 2.2.5.6

Divide each term in $- x = - 2$ by $- 1$ and simplify.

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Step 2.2.5.6.1

Divide each term in $- x = - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 2}{- 1}$

Step 2.2.5.6.2

Simplify the left side.

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Step 2.2.5.6.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 2}{- 1}$

Step 2.2.5.6.2.2

Divide $x$ by $1$ .

$x = \frac{- 2}{- 1}$

Step 2.2.5.6.3

Simplify the right side.

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Step 2.2.5.6.3.1

Divide $- 2$ by $- 1$ .

$x = 2$

Step 2.2.5.7

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

$x = 0, 2$