Trigonometry Examples

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

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$- \frac{x}{3} = 0$

Step 1.2

Set the numerator equal to zero.

$x = 0$

Step 1.3

The vertical asymptote occurs at $x = 0$ .

Vertical Asymptote: $x = 0$

Step 2

Find the point at $x = - 1$ .

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Step 2.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = - \log_{3} (- \frac{1}{3} \cdot - 1)$

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $- \frac{1}{3} \cdot - 1$ .

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Step 2.2.1.1

Multiply $- 1$ by $- 1$ .

$f (- 1) = - \log_{3} (1 (\frac{1}{3}))$

Step 2.2.1.2

Multiply $\frac{1}{3}$ by $1$ .

$f (- 1) = - \log_{3} (\frac{1}{3})$

Step 2.2.2

Logarithm base $3$ of $\frac{1}{3}$ is $- 1$ .

$f (- 1) = 1$

Step 2.2.3

The final answer is $1$ .

$1$

Step 2.3

Convert $1$ to decimal.

$= 1$

Step 3

Find the point at $x = - 2$ .

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Step 3.1

Replace the variable $x$ with $- 2$ in the expression.

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

Step 3.2

Simplify the result.

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Step 3.2.1

Multiply $- \frac{1}{3} \cdot - 2$ .

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Step 3.2.1.1

Multiply $- 2$ by $- 1$ .

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

Step 3.2.1.2

Combine $2$ and $\frac{1}{3}$ .

$f (- 2) = - \log_{3} (\frac{2}{3})$

Step 3.2.2

Rewrite $\frac{2}{3}$ as $2 \cdot 3^{- 1}$ .

$f (- 2) = - \log_{3} (2 \cdot 3^{- 1})$

Step 3.2.3

Rewrite $\log_{3} (2 \cdot 3^{- 1})$ as $\log_{3} (2) + \log_{3} (3^{- 1})$ .

$f (- 2) = - (\log_{3} (2) + \log_{3} (3^{- 1}))$

Step 3.2.4

Use logarithm rules to move $- 1$ out of the exponent.

$f (- 2) = - (\log_{3} (2) - \log_{3} (3))$

Step 3.2.5

Logarithm base $3$ of $3$ is $1$ .

$f (- 2) = - (\log_{3} (2) - 1 \cdot 1)$

Step 3.2.6

Multiply $- 1$ by $1$ .

$f (- 2) = - (\log_{3} (2) - 1)$

Step 3.2.7

Apply the distributive property.

$f (- 2) = - \log_{3} (2) + 1$

Step 3.2.8

The final answer is $- \log_{3} (2) + 1$ .

$- \log_{3} (2) + 1$

Step 3.3

Convert $- \log_{3} (2) + 1$ to decimal.

$= 0.36907024$

Step 4

Find the point at $x = - 3$ .

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Step 4.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = - \log_{3} (- \frac{1}{3} \cdot - 3)$

Step 4.2

Simplify the result.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$f (- 3) = - \log_{3} (\frac{- 1}{3} \cdot - 3)$

Step 4.2.1.2

Factor $3$ out of $- 3$ .

$f (- 3) = - \log_{3} (\frac{- 1}{3} \cdot (3 (- 1)))$

Step 4.2.1.3

Cancel the common factor.

$f (- 3) = - \log_{3} (\frac{- 1}{3} \cdot (3 \cdot - 1))$

Step 4.2.1.4

Rewrite the expression.

$f (- 3) = - \log_{3} (- 1 \cdot - 1)$

Step 4.2.2

Multiply $- 1$ by $- 1$ .

$f (- 3) = - \log_{3} (1)$

Step 4.2.3

Logarithm base $3$ of $1$ is $0$ .

$f (- 3) = - 0$

Step 4.2.4

Multiply $- 1$ by $0$ .

$f (- 3) = 0$

Step 4.2.5

The final answer is $0$ .

$0$

Step 4.3

Convert $0$ to decimal.

$= 0$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(- 1, 1), (- 2, 0.36907024), (- 3, 0)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ - 3 & 0 \\ - 2 & 0.369 \\ - 1 & 1 \end{array}$

Step 6