Trigonometry Examples

$\log_{2} (15) + \log_{2} (14) - \log_{2} (105) = \log (x)$

Step 1

Subtract $\log (x)$ from both sides of the equation.

$\log_{2} (15) + \log_{2} (14) - \log_{2} (105) - \log (x) = 0$

Step 2

Simplify $\log_{2} (15) + \log_{2} (14) - \log_{2} (105) - \log (x)$ .

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Step 2.1

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

$\log_{2} (15 \cdot 14) - \log_{2} (105) - \log (x) = 0$

Step 2.2

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

$\log_{2} (\frac{15 \cdot 14}{105}) - \log (x) = 0$

Step 2.3

Simplify each term.

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Step 2.3.1

Cancel the common factor of $15$ and $105$ .

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Step 2.3.1.1

Factor $15$ out of $15 \cdot 14$ .

$\log_{2} (\frac{15 (14)}{105}) - \log (x) = 0$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $15$ out of $105$ .

$\log_{2} (\frac{15 \cdot 14}{15 \cdot 7}) - \log (x) = 0$

Step 2.3.1.2.2

Cancel the common factor.

$\log_{2} (\frac{15 \cdot 14}{15 \cdot 7}) - \log (x) = 0$

Step 2.3.1.2.3

Rewrite the expression.

$\log_{2} (\frac{14}{7}) - \log (x) = 0$

Step 2.3.2

Cancel the common factor of $14$ and $7$ .

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Step 2.3.2.1

Factor $7$ out of $14$ .

$\log_{2} (\frac{7 \cdot 2}{7}) - \log (x) = 0$

Step 2.3.2.2

Cancel the common factors.

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Step 2.3.2.2.1

Factor $7$ out of $7$ .

$\log_{2} (\frac{7 \cdot 2}{7 (1)}) - \log (x) = 0$

Step 2.3.2.2.2

Cancel the common factor.

$\log_{2} (\frac{7 \cdot 2}{7 \cdot 1}) - \log (x) = 0$

Step 2.3.2.2.3

Rewrite the expression.

$\log_{2} (\frac{2}{1}) - \log (x) = 0$

Step 2.3.2.2.4

Divide $2$ by $1$ .

$\log_{2} (2) - \log (x) = 0$

Step 2.3.3

Logarithm base $2$ of $2$ is $1$ .

$1 - \log (x) = 0$

Step 3

Set the argument in $\log (x)$ less than or equal to $0$ to find where the expression is undefined.

$x \leq 0$

Step 4

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 $0$ .

$x \leq 0$

$(- \infty, 0]$

Step 5