Precalculus Examples

$\log_{5} (- 45 x) - 2 \log_{5} (2 - x) = 1 + \log_{5} (- x)$

Step 1

Reorder $2$ and $- x$ .

$\log_{5} (- 45 x) - 2 \log_{5} (- x + 2) = 1 + \log_{5} (- x)$

Step 2

Move all the terms containing a logarithm to the left side of the equation.

$\log_{5} (- 45 x) - 2 \log_{5} (- x + 2) - \log_{5} (- x) = 1$

Step 3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{5} (\frac{- 45 x}{- x}) - 2 \log_{5} (- x + 2) = 1$

Step 4

Simplify each term.

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Step 4.1

Cancel the common factor of $x$ .

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Step 4.1.1

Cancel the common factor.

$\log_{5} (\frac{- 45 x}{- x}) - 2 \log_{5} (- x + 2) = 1$

Step 4.1.2

Rewrite the expression.

$\log_{5} (\frac{- 45}{- 1}) - 2 \log_{5} (- x + 2) = 1$

Step 4.1.3

Move the negative one from the denominator of $\frac{- 45}{- 1}$ .

$\log_{5} (- 1 \cdot - 45) - 2 \log_{5} (- x + 2) = 1$

Step 4.2

Rewrite $- 1 \cdot - 45$ as $- - 45$ .

$\log_{5} (45) - 2 \log_{5} (- x + 2) = 1$

Step 4.3

Multiply $- 1$ by $- 45$ .

$\log_{5} (45) - 2 \log_{5} (- x + 2) = 1$

Step 5

Simplify the left side.

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Step 5.1

Simplify $\log_{5} (45) - 2 \log_{5} (- x + 2)$ .

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Step 5.1.1

Simplify $- 2 \log_{5} (- x + 2)$ by moving $2$ inside the logarithm.

$\log_{5} (45) - \log_{5} ({(- x + 2)}^{2}) = 1$

Step 5.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{5} (\frac{45}{{(- x + 2)}^{2}}) = 1$

Step 6

Rewrite $\log_{5} (\frac{45}{{(- x + 2)}^{2}}) = 1$ 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 $b^{y} = x$ .

$5^{1} = \frac{45}{{(- x + 2)}^{2}}$

Step 7

Solve for $x$ .

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Step 7.1

Rewrite the equation as $\frac{45}{{(- x + 2)}^{2}} = 5^{1}$ .

$\frac{45}{{(- x + 2)}^{2}} = 5$

Step 7.2

Find the LCD of the terms in the equation.

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Step 7.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

${(- x + 2)}^{2}, 1$

Step 7.2.2

The LCM of one and any expression is the expression.

${(- x + 2)}^{2}$

Step 7.3

Multiply each term in $\frac{45}{{(- x + 2)}^{2}} = 5$ by ${(- x + 2)}^{2}$ to eliminate the fractions.

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Step 7.3.1

Multiply each term in $\frac{45}{{(- x + 2)}^{2}} = 5$ by ${(- x + 2)}^{2}$ .

$\frac{45}{{(- x + 2)}^{2}} {(- x + 2)}^{2} = 5 {(- x + 2)}^{2}$

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Cancel the common factor of ${(- x + 2)}^{2}$ .

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Step 7.3.2.1.1

Cancel the common factor.

$\frac{45}{{(- x + 2)}^{2}} {(- x + 2)}^{2} = 5 {(- x + 2)}^{2}$

Step 7.3.2.1.2

Rewrite the expression.

$45 = 5 {(- x + 2)}^{2}$

Step 7.4

Solve the equation.

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Step 7.4.1

Rewrite the equation as $5 {(- x + 2)}^{2} = 45$ .

$5 {(- x + 2)}^{2} = 45$

Step 7.4.2

Divide each term in $5 {(- x + 2)}^{2} = 45$ by $5$ and simplify.

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Step 7.4.2.1

Divide each term in $5 {(- x + 2)}^{2} = 45$ by $5$ .

$\frac{5 {(- x + 2)}^{2}}{5} = \frac{45}{5}$

Step 7.4.2.2

Simplify the left side.

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Step 7.4.2.2.1

Cancel the common factor of $5$ .

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Step 7.4.2.2.1.1

Cancel the common factor.

$\frac{5 {(- x + 2)}^{2}}{5} = \frac{45}{5}$

Step 7.4.2.2.1.2

Divide ${(- x + 2)}^{2}$ by $1$ .

${(- x + 2)}^{2} = \frac{45}{5}$

Step 7.4.2.3

Simplify the right side.

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Step 7.4.2.3.1

Divide $45$ by $5$ .

${(- x + 2)}^{2} = 9$

Step 7.4.3

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

$- x + 2 = \pm \sqrt{9}$

Step 7.4.4

Simplify $\pm \sqrt{9}$ .

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Step 7.4.4.1

Rewrite $9$ as $3^{2}$ .

$- x + 2 = \pm \sqrt{3^{2}}$

Step 7.4.4.2

Pull terms out from under the radical, assuming positive real numbers.

$- x + 2 = \pm 3$

Step 7.4.5

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

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Step 7.4.5.1

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

$- x + 2 = 3$

Step 7.4.5.2

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

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Step 7.4.5.2.1

Subtract $2$ from both sides of the equation.

$- x = 3 - 2$

Step 7.4.5.2.2

Subtract $2$ from $3$ .

$- x = 1$

Step 7.4.5.3

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

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Step 7.4.5.3.1

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

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

Step 7.4.5.3.2

Simplify the left side.

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Step 7.4.5.3.2.1

Dividing two negative values results in a positive value.

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

Step 7.4.5.3.2.2

Divide $x$ by $1$ .

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

Step 7.4.5.3.3

Simplify the right side.

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Step 7.4.5.3.3.1

Divide $1$ by $- 1$ .

$x = - 1$

Step 7.4.5.4

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

$- x + 2 = - 3$

Step 7.4.5.5

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

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Step 7.4.5.5.1

Subtract $2$ from both sides of the equation.

$- x = - 3 - 2$

Step 7.4.5.5.2

Subtract $2$ from $- 3$ .

$- x = - 5$

Step 7.4.5.6

Divide each term in $- x = - 5$ by $- 1$ and simplify.

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Step 7.4.5.6.1

Divide each term in $- x = - 5$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 5}{- 1}$

Step 7.4.5.6.2

Simplify the left side.

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Step 7.4.5.6.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 5}{- 1}$

Step 7.4.5.6.2.2

Divide $x$ by $1$ .

$x = \frac{- 5}{- 1}$

Step 7.4.5.6.3

Simplify the right side.

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Step 7.4.5.6.3.1

Divide $- 5$ by $- 1$ .

$x = 5$

Step 7.4.5.7

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

$x = - 1, 5$

Step 8

Exclude the solutions that do not make $\log_{5} (- 45 x) - 2 \log_{5} (2 - x) = 1 + \log_{5} (- x)$ true.

$x = - 1$