Precalculus Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Simplify the expression.

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Step 1.3.1

Multiply $x$ by $x$ .

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

Step 1.3.2

Move $- 3$ to the left of $x$ .

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

Step 2

Rewrite $\log (x^{2} - 3 x) = 2$ 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 $b^{y} = x$ .

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

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $x^{2} - 3 x = 10^{2}$ .

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

Step 3.2

Raise $10$ to the power of $2$ .

$x^{2} - 3 x = 100$

Step 3.3

Subtract $100$ from both sides of the equation.

$x^{2} - 3 x - 100 = 0$

Step 3.4

Use the quadratic formula to find the solutions.

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

Step 3.5

Substitute the values $a = 1$ , $b = - 3$ , and $c = - 100$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (1 \cdot - 100)}}{2 \cdot 1}$

Step 3.6

Simplify.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 100}}{2 \cdot 1}$

Step 3.6.1.2

Multiply $- 4 \cdot 1 \cdot - 100$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 100}}{2 \cdot 1}$

Step 3.6.1.2.2

Multiply $- 4$ by $- 100$ .

$x = \frac{3 \pm \sqrt{9 + 400}}{2 \cdot 1}$

Step 3.6.1.3

Add $9$ and $400$ .

$x = \frac{3 \pm \sqrt{409}}{2 \cdot 1}$

Step 3.6.2

Multiply $2$ by $1$ .

$x = \frac{3 \pm \sqrt{409}}{2}$

Step 3.7

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{409}}{2}, \frac{3 - \sqrt{409}}{2}$

Step 4

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

$x = \frac{3 + \sqrt{409}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + \sqrt{409}}{2}$

Decimal Form:

$x = 11.61187420 \dots$