Precalculus Examples

$\ln (x + 8) + \ln (x - 5) = 2 \ln (x)$

Step 1

Simplify the left side.

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Step 1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln ((x + 8) (x - 5)) = 2 \ln (x)$

Step 1.2

Expand $(x + 8) (x - 5)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\ln (x (x - 5) + 8 (x - 5)) = 2 \ln (x)$

Step 1.2.2

Apply the distributive property.

$\ln (x \cdot x + x \cdot - 5 + 8 (x - 5)) = 2 \ln (x)$

Step 1.2.3

Apply the distributive property.

$\ln (x \cdot x + x \cdot - 5 + 8 x + 8 \cdot - 5) = 2 \ln (x)$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

$\ln (x^{2} + x \cdot - 5 + 8 x + 8 \cdot - 5) = 2 \ln (x)$

Step 1.3.1.2

Move $- 5$ to the left of $x$ .

$\ln (x^{2} - 5 \cdot x + 8 x + 8 \cdot - 5) = 2 \ln (x)$

Step 1.3.1.3

Multiply $8$ by $- 5$ .

$\ln (x^{2} - 5 x + 8 x - 40) = 2 \ln (x)$

Step 1.3.2

Add $- 5 x$ and $8 x$ .

$\ln (x^{2} + 3 x - 40) = 2 \ln (x)$

Step 2

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

$\ln (x^{2} + 3 x - 40) = \ln (x^{2})$

Step 3

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$x^{2} + 3 x - 40 = x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Move all terms containing $x$ to the left side of the equation.

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Step 4.1.1

Subtract $x^{2}$ from both sides of the equation.

$x^{2} + 3 x - 40 - x^{2} = 0$

Step 4.1.2

Combine the opposite terms in $x^{2} + 3 x - 40 - x^{2}$ .

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Step 4.1.2.1

Subtract $x^{2}$ from $x^{2}$ .

$3 x - 40 + 0 = 0$

Step 4.1.2.2

Add $3 x - 40$ and $0$ .

$3 x - 40 = 0$

Step 4.2

Add $40$ to both sides of the equation.

$3 x = 40$

Step 4.3

Divide each term in $3 x = 40$ by $3$ and simplify.

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Step 4.3.1

Divide each term in $3 x = 40$ by $3$ .

$\frac{3 x}{3} = \frac{40}{3}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $3$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{40}{3}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{40}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{40}{3}$

Decimal Form:

$x = 13.\overline{3}$

Mixed Number Form:

$x = 13 \frac{1}{3}$