Precalculus Examples

$\log (x - 2) + \log (x + 5) = 2 \log (3)$

Step 1

Simplify the left side.

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Step 1.1

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

$\log ((x - 2) (x + 5)) = 2 \log (3)$

Step 1.2

Expand $(x - 2) (x + 5)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\log (x (x + 5) - 2 (x + 5)) = 2 \log (3)$

Step 1.2.2

Apply the distributive property.

$\log (x \cdot x + x \cdot 5 - 2 (x + 5)) = 2 \log (3)$

Step 1.2.3

Apply the distributive property.

$\log (x \cdot x + x \cdot 5 - 2 x - 2 \cdot 5) = 2 \log (3)$

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 $x$ .

$\log (x^{2} + x \cdot 5 - 2 x - 2 \cdot 5) = 2 \log (3)$

Step 1.3.1.2

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

$\log (x^{2} + 5 \cdot x - 2 x - 2 \cdot 5) = 2 \log (3)$

Step 1.3.1.3

Multiply $- 2$ by $5$ .

$\log (x^{2} + 5 x - 2 x - 10) = 2 \log (3)$

Step 1.3.2

Subtract $2 x$ from $5 x$ .

$\log (x^{2} + 3 x - 10) = 2 \log (3)$

Step 2

Simplify $2 \log (3)$ by moving $2$ inside the logarithm.

$\log (x^{2} + 3 x - 10) = \log (3^{2})$

Step 3

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

$x^{2} + 3 x - 10 = 3^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Raise $3$ to the power of $2$ .

$x^{2} + 3 x - 10 = 9$

Step 4.2

Move all terms to the left side of the equation and simplify.

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Step 4.2.1

Subtract $9$ from both sides of the equation.

$x^{2} + 3 x - 10 - 9 = 0$

Step 4.2.2

Subtract $9$ from $- 10$ .

$x^{2} + 3 x - 19 = 0$

Step 4.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.4

Substitute the values $a = 1$ , $b = 3$ , and $c = - 19$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot - 19)}}{2 \cdot 1}$

Step 4.5

Simplify.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 19}}{2 \cdot 1}$

Step 4.5.1.2

Multiply $- 4 \cdot 1 \cdot - 19$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 19}}{2 \cdot 1}$

Step 4.5.1.2.2

Multiply $- 4$ by $- 19$ .

$x = \frac{- 3 \pm \sqrt{9 + 76}}{2 \cdot 1}$

Step 4.5.1.3

Add $9$ and $76$ .

$x = \frac{- 3 \pm \sqrt{85}}{2 \cdot 1}$

Step 4.5.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{85}}{2}$

Step 4.6

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{85}}{2}, - \frac{3 + \sqrt{85}}{2}$

Step 5

Exclude the solutions that do not make $\log (x - 2) + \log (x + 5) = 2 \log (3)$ true.

$x = - \frac{3 - \sqrt{85}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3 - \sqrt{85}}{2}$

Decimal Form:

$x = 3.10977222 \dots$