Trigonometry Examples

$\log_{2} (x + 8) = \log_{2} (3) + \log_{2} (5)$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\log_{2} (3)$ from both sides of the equation.

$\log_{2} (x + 8) - \log_{2} (3) = \log_{2} (5)$

Step 1.2

Subtract $\log_{2} (5)$ from both sides of the equation.

$\log_{2} (x + 8) - \log_{2} (3) - \log_{2} (5) = 0$

Step 2

Simplify $\log_{2} (x + 8) - \log_{2} (3) - \log_{2} (5)$ .

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Step 2.1

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

$\log_{2} (\frac{x + 8}{3}) - \log_{2} (5) = 0$

Step 2.2

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

$\log_{2} (\frac{\frac{x + 8}{3}}{5}) = 0$

Step 2.3

Multiply the numerator by the reciprocal of the denominator.

$\log_{2} (\frac{x + 8}{3} \cdot \frac{1}{5}) = 0$

Step 2.4

Multiply $\frac{x + 8}{3} \cdot \frac{1}{5}$ .

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Step 2.4.1

Multiply $\frac{x + 8}{3}$ by $\frac{1}{5}$ .

$\log_{2} (\frac{x + 8}{3 \cdot 5}) = 0$

Step 2.4.2

Multiply $3$ by $5$ .

$\log_{2} (\frac{x + 8}{15}) = 0$

Step 3

Set the argument in $\log_{2} (\frac{x + 8}{15})$ less than or equal to $0$ to find where the expression is undefined.

$\frac{x + 8}{15} \leq 0$

Step 4

Solve for $x$ .

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Step 4.1

Multiply both sides by $15$ .

$\frac{x + 8}{15} \cdot 15 \leq 0 \cdot 15$

Step 4.2

Simplify.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Cancel the common factor of $15$ .

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Step 4.2.1.1.1

Cancel the common factor.

$\frac{x + 8}{15} \cdot 15 \leq 0 \cdot 15$

Step 4.2.1.1.2

Rewrite the expression.

$x + 8 \leq 0 \cdot 15$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Multiply $0$ by $15$ .

$x + 8 \leq 0$

Step 4.3

Subtract $8$ from both sides of the inequality.

$x \leq - 8$

Step 5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq - 8$

$(- \infty, - 8]$

Step 6