Trigonometry Examples

$\frac{1}{3} \cdot \ln (\frac{x - 1}{2})$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$\frac{{(x - 1)}^{\frac{1}{3}}}{2^{\frac{1}{3}}} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Set the numerator equal to zero.

${(x - 1)}^{\frac{1}{3}} = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 1.2.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.3

The vertical asymptote occurs at $x = 1$ .

Vertical Asymptote: $x = 1$

Step 2

Find the point at $x = 3$ .

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Step 2.1

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

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

Step 2.2

Simplify the result.

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Step 2.2.1

Subtract $1$ from $3$ .

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

Step 2.2.2

Divide $2$ by $2$ .

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

Step 2.2.3

The natural logarithm of $1$ is $0$ .

$f (3) = \frac{1}{3} \cdot 0$

Step 2.2.4

Multiply $\frac{1}{3}$ by $0$ .

$f (3) = 0$

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.

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

Step 3.2

Simplify the result.

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Step 3.2.1

Subtract $1$ from $2$ .

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

Step 3.2.2

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

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

Step 3.2.3

Apply the product rule to $\frac{1}{2}$ .

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

Step 3.2.4

One to any power is one.

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

Step 3.2.5

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

$\ln (\frac{1}{2^{\frac{1}{3}}})$

Step 3.3

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

$y = - 0.23104906$

Step 4

Find the point at $x = 4$ .

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Step 4.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = \frac{1}{3} \cdot \ln (\frac{(4) - 1}{2})$

Step 4.2

Simplify the result.

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Step 4.2.1

Subtract $1$ from $4$ .

$f (4) = \frac{1}{3} \cdot \ln (\frac{3}{2})$

Step 4.2.2

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

$f (4) = \ln ({(\frac{3}{2})}^{\frac{1}{3}})$

Step 4.2.3

Apply the product rule to $\frac{3}{2}$ .

$f (4) = \ln (\frac{3^{\frac{1}{3}}}{2^{\frac{1}{3}}})$

Step 4.2.4

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

$\ln (\frac{3^{\frac{1}{3}}}{2^{\frac{1}{3}}})$

Step 4.3

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

$y = 0.13515503$

Step 5

The log function can be graphed using the vertical asymptote at $x = 1$ and the points $(3, 0), (2, - 0.23104906), (4, 0.13515503)$ .

Vertical Asymptote: $x = 1$

$\begin{array}{c} x & y \\ 2 & - 0.231 \\ 3 & 0 \\ 4 & 0.135 \end{array}$

Step 6