Trigonometry Examples

$\log_{2} (9 x) - \log_{2} (3) = 3$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{2} (\frac{9 x}{3}) = 3$

Step 1.2

Cancel the common factor of $9$ and $3$ .

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Step 1.2.1

Factor $3$ out of $9 x$ .

$\log_{2} (\frac{3 (3 x)}{3}) = 3$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $3$ out of $3$ .

$\log_{2} (\frac{3 (3 x)}{3 (1)}) = 3$

Step 1.2.2.2

Cancel the common factor.

$\log_{2} (\frac{3 (3 x)}{3 \cdot 1}) = 3$

Step 1.2.2.3

Rewrite the expression.

$\log_{2} (\frac{3 x}{1}) = 3$

Step 1.2.2.4

Divide $3 x$ by $1$ .

$\log_{2} (3 x) = 3$

Step 2

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

$2^{3} = 3 x$

Step 3

Solve for $x$ .

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Step 3.1

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

$3 x = 2^{3}$

Step 3.2

Divide each term in $3 x = 2^{3}$ by $3$ and simplify.

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Step 3.2.1

Divide each term in $3 x = 2^{3}$ by $3$ .

$\frac{3 x}{3} = \frac{2^{3}}{3}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{2^{3}}{3}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2^{3}}{3}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

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

$x = \frac{8}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{8}{3}$

Decimal Form:

$x = 2.\overline{6}$

Mixed Number Form:

$x = 2 \frac{2}{3}$