Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.1

Move $x$ .

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

Step 1.3.2

Multiply $x$ by $x$ .

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

Step 1.4

Multiply $- 2$ by $5$ .

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

Step 2

Rewrite $\log (5 x^{2} - 10 x) = 2$ 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 $b^{y} = x$ .

$10^{2} = 5 x^{2} - 10 x$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $5 x^{2} - 10 x = 10^{2}$ .

$5 x^{2} - 10 x = 10^{2}$

Step 3.2

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

$5 x^{2} - 10 x = 100$

Step 3.3

Subtract $100$ from both sides of the equation.

$5 x^{2} - 10 x - 100 = 0$

Step 3.4

Factor $5$ out of $5 x^{2} - 10 x - 100$ .

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Step 3.4.1

Factor $5$ out of $5 x^{2}$ .

$5 (x^{2}) - 10 x - 100 = 0$

Step 3.4.2

Factor $5$ out of $- 10 x$ .

$5 (x^{2}) + 5 (- 2 x) - 100 = 0$

Step 3.4.3

Factor $5$ out of $- 100$ .

$5 x^{2} + 5 (- 2 x) + 5 \cdot - 20 = 0$

Step 3.4.4

Factor $5$ out of $5 x^{2} + 5 (- 2 x)$ .

$5 (x^{2} - 2 x) + 5 \cdot - 20 = 0$

Step 3.4.5

Factor $5$ out of $5 (x^{2} - 2 x) + 5 \cdot - 20$ .

$5 (x^{2} - 2 x - 20) = 0$

Step 3.5

Divide each term in $5 (x^{2} - 2 x - 20) = 0$ by $5$ and simplify.

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Step 3.5.1

Divide each term in $5 (x^{2} - 2 x - 20) = 0$ by $5$ .

$\frac{5 (x^{2} - 2 x - 20)}{5} = \frac{0}{5}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $5$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{5 (x^{2} - 2 x - 20)}{5} = \frac{0}{5}$

Step 3.5.2.1.2

Divide $x^{2} - 2 x - 20$ by $1$ .

$x^{2} - 2 x - 20 = \frac{0}{5}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Divide $0$ by $5$ .

$x^{2} - 2 x - 20 = 0$

Step 3.6

Use the quadratic formula to find the solutions.

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

Step 3.7

Substitute the values $a = 1$ , $b = - 2$ , and $c = - 20$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot - 20)}}{2 \cdot 1}$

Step 3.8

Simplify.

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Step 3.8.1

Simplify the numerator.

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Step 3.8.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 20}}{2 \cdot 1}$

Step 3.8.1.2

Multiply $- 4 \cdot 1 \cdot - 20$ .

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Step 3.8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 20}}{2 \cdot 1}$

Step 3.8.1.2.2

Multiply $- 4$ by $- 20$ .

$x = \frac{2 \pm \sqrt{4 + 80}}{2 \cdot 1}$

Step 3.8.1.3

Add $4$ and $80$ .

$x = \frac{2 \pm \sqrt{84}}{2 \cdot 1}$

Step 3.8.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 3.8.1.4.1

Factor $4$ out of $84$ .

$x = \frac{2 \pm \sqrt{4 (21)}}{2 \cdot 1}$

Step 3.8.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 1}$

Step 3.8.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{21}}{2 \cdot 1}$

Step 3.8.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 \sqrt{21}}{2}$

Step 3.8.3

Simplify $\frac{2 \pm 2 \sqrt{21}}{2}$ .

$x = 1 \pm \sqrt{21}$

Step 3.9

The final answer is the combination of both solutions.

$x = 1 + \sqrt{21}, 1 - \sqrt{21}$

Step 4

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

$x = 1 + \sqrt{21}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 1 + \sqrt{21}$

Decimal Form:

$x = 5.58257569 \dots$