Trigonometry Examples

$y = \frac{4 \log_{3} (x)}{11 x^{3}}$

Step 1

Find the asymptotes.

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Step 1.1

Find where the expression $\frac{\log_{3} (x^{4})}{11 x^{3}}$ is undefined.

$x = 0$

Step 1.2

Ignoring the logarithm, consider the rational function $R (x) = \frac{a x^{n}}{b x^{m}}$ where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

1. If $n < m$ , then the x-axis, $y = 0$ , is the horizontal asymptote.

2. If $n = m$ , then the horizontal asymptote is the line $y = \frac{a}{b}$ .

3. If $n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 1.3

Find $n$ and $m$ .

$n = 0$

$m = 3$

Step 1.4

Since $n < m$ , the x-axis, $y = 0$ , is the horizontal asymptote.

$y = 0$

Step 1.5

No oblique asymptotes are present for logarithmic and trigonometric functions.

No Oblique Asymptotes

Step 1.6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = 0$

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = 0$

Step 2

Find the point at $x = 1$ .

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Step 2.1

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

$f (1) = \frac{4 \log_{3} (1)}{11 {(1)}^{3}}$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify $4 \log_{3} (1)$ by moving $4$ inside the logarithm.

$f (1) = \frac{\log_{3} (1^{4})}{11 \cdot 1^{3}}$

Step 2.2.2

One to any power is one.

$f (1) = \frac{\log_{3} (1^{4})}{11 \cdot 1}$

Step 2.2.3

Simplify the numerator.

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Step 2.2.3.1

One to any power is one.

$f (1) = \frac{\log_{3} (1)}{11 \cdot 1}$

Step 2.2.3.2

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

$f (1) = \frac{0}{11 \cdot 1}$

Step 2.2.4

Simplify the expression.

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Step 2.2.4.1

Multiply $11$ by $1$ .

$f (1) = \frac{0}{11}$

Step 2.2.4.2

Divide $0$ by $11$ .

$f (1) = 0$

Step 2.2.5

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = \frac{4 \log_{3} (2)}{11 {(2)}^{3}}$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify $4 \log_{3} (2)$ by moving $4$ inside the logarithm.

$f (2) = \frac{\log_{3} (2^{4})}{11 \cdot 2^{3}}$

Step 3.2.2

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

$f (2) = \frac{\log_{3} (2^{4})}{11 \cdot 8}$

Step 3.2.3

Raise $2$ to the power of $4$ .

$f (2) = \frac{\log_{3} (16)}{11 \cdot 8}$

Step 3.2.4

Multiply $11$ by $8$ .

$f (2) = \frac{\log_{3} (16)}{88}$

Step 3.2.5

Rewrite $\log_{3} (16)$ as $\log_{3} (2^{4})$ .

$f (2) = \frac{\log_{3} (2^{4})}{88}$

Step 3.2.6

Expand $\log_{3} (2^{4})$ by moving $4$ outside the logarithm.

$f (2) = \frac{4 \log_{3} (2)}{88}$

Step 3.2.7

Cancel the common factor of $4$ and $88$ .

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Step 3.2.7.1

Factor $4$ out of $4 \log_{3} (2)$ .

$f (2) = \frac{4 (\log_{3} (2))}{88}$

Step 3.2.7.2

Cancel the common factors.

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Step 3.2.7.2.1

Factor $4$ out of $88$ .

$f (2) = \frac{4 \log_{3} (2)}{4 \cdot 22}$

Step 3.2.7.2.2

Cancel the common factor.

$f (2) = \frac{4 \log_{3} (2)}{4 \cdot 22}$

Step 3.2.7.2.3

Rewrite the expression.

$f (2) = \frac{\log_{3} (2)}{22}$

Step 3.2.8

Rewrite $\frac{\log_{3} (2)}{22}$ as $\frac{1}{22} \log_{3} (2)$ .

$f (2) = \frac{1}{22} \cdot \log_{3} (2)$

Step 3.2.9

Simplify $\frac{1}{22} \log_{3} (2)$ by moving $\frac{1}{22}$ inside the logarithm.

$f (2) = \log_{3} (2^{\frac{1}{22}})$

Step 3.2.10

The final answer is $\log_{3} (2^{\frac{1}{22}})$ .

$\log_{3} (2^{\frac{1}{22}})$

Step 3.3

Convert $\log_{3} (2^{\frac{1}{22}})$ to decimal.

$y = 0.02867862$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

$f (3) = \frac{4 \log_{3} (3)}{11 {(3)}^{3}}$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify $4 \log_{3} (3)$ by moving $4$ inside the logarithm.

$f (3) = \frac{\log_{3} (3^{4})}{11 \cdot 3^{3}}$

Step 4.2.2

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

$f (3) = \frac{\log_{3} (3^{4})}{11 \cdot 27}$

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Use logarithm rules to move $4$ out of the exponent.

$f (3) = \frac{4 \log_{3} (3)}{11 \cdot 27}$

Step 4.2.3.2

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

$f (3) = \frac{4 \cdot 1}{11 \cdot 27}$

Step 4.2.3.3

Multiply $4$ by $1$ .

$f (3) = \frac{4}{11 \cdot 27}$

Step 4.2.4

Multiply $11$ by $27$ .

$f (3) = \frac{4}{297}$

Step 4.2.5

The final answer is $\frac{4}{297}$ .

$\frac{4}{297}$

Step 4.3

Convert $\frac{4}{297}$ to decimal.

$y = 0.\overline{013468}$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0), (2, 0.02867862), (3, 0.\overline{013468})$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 0.029 \\ 3 & 0.013 \end{array}$

Step 6