Calculus Examples

$f (x) = \frac{10 \ln (x - 1)}{x}$

Step 1

Find the asymptotes.

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Step 1.1

Find where the expression $\frac{\ln ({(x - 1)}^{10})}{x}$ is undefined.

$x = 0, x = 1$

Step 1.2

Since $\frac{\ln ({(x - 1)}^{10})}{x}$ $\to$ $- \infty$ as $x$ $\to$ $1$ from the left and $\frac{\ln ({(x - 1)}^{10})}{x}$ $\to$ $- \infty$ as $x$ $\to$ $1$ from the right, then $x = 1$ is a vertical asymptote.

$x = 1$

Step 1.3

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 1.4

Find $n$ and $m$ .

$n = 0$

$m = 1$

Step 1.5

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

$y = 0$

Step 1.6

No oblique asymptotes are present for logarithmic and trigonometric functions.

No Oblique Asymptotes

Step 1.7

This is the set of all asymptotes.

Vertical Asymptotes: $x = 1$

Horizontal Asymptotes: $y = 0$

Vertical Asymptotes: $x = 1$

Horizontal Asymptotes: $y = 0$

Step 2

Find the point at $x = 2$ .

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Step 2.1

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

$f (2) = \frac{10 \ln ((2) - 1)}{2}$

Step 2.2

Simplify the result.

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Step 2.2.1

Cancel the common factor of $10$ and $2$ .

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Step 2.2.1.1

Factor $2$ out of $10 \ln ((2) - 1)$ .

$f (2) = \frac{2 (5 \ln ((2) - 1))}{2}$

Step 2.2.1.2

Cancel the common factors.

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Step 2.2.1.2.1

Factor $2$ out of $2$ .

$f (2) = \frac{2 (5 \ln ((2) - 1))}{2 (1)}$

Step 2.2.1.2.2

Cancel the common factor.

$f (2) = \frac{2 (5 \ln ((2) - 1))}{2 \cdot 1}$

Step 2.2.1.2.3

Rewrite the expression.

$f (2) = \frac{5 \ln ((2) - 1)}{1}$

Step 2.2.1.2.4

Divide $5 \ln ((2) - 1)$ by $1$ .

$f (2) = 5 \ln ((2) - 1)$

Step 2.2.2

Subtract $1$ from $2$ .

$f (2) = 5 \ln (1)$

Step 2.2.3

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

$f (2) = 5 \cdot 0$

Step 2.2.4

Multiply $5$ by $0$ .

$f (2) = 0$

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 $x = 3$ .

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Step 3.1

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

$f (3) = \frac{10 \ln ((3) - 1)}{3}$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify $10 \ln (3 - 1)$ by moving $10$ inside the logarithm.

$f (3) = \frac{\ln ({(3 - 1)}^{10})}{3}$

Step 3.2.2

Simplify the numerator.

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Step 3.2.2.1

Subtract $1$ from $3$ .

$f (3) = \frac{\ln (2^{10})}{3}$

Step 3.2.2.2

Raise $2$ to the power of $10$ .

$f (3) = \frac{\ln (1024)}{3}$

Step 3.2.3

Rewrite $\frac{\ln (1024)}{3}$ as $\frac{1}{3} \ln (1024)$ .

$f (3) = \frac{1}{3} \cdot \ln (1024)$

Step 3.2.4

Simplify $\frac{1}{3} \ln (1024)$ by moving $\frac{1}{3}$ inside the logarithm.

$f (3) = \ln (1024^{\frac{1}{3}})$

Step 3.2.5

The final answer is $\ln (1024^{\frac{1}{3}})$ .

$\ln (1024^{\frac{1}{3}})$

Step 3.3

Convert $\ln (1024^{\frac{1}{3}})$ to decimal.

$y = 2.3104906$

Step 4

Find the point at $x = 4$ .

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Step 4.1

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

$f (4) = \frac{10 \ln ((4) - 1)}{4}$

Step 4.2

Simplify the result.

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Step 4.2.1

Cancel the common factor of $10$ and $4$ .

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Step 4.2.1.1

Factor $2$ out of $10 \ln ((4) - 1)$ .

$f (4) = \frac{2 (5 \ln ((4) - 1))}{4}$

Step 4.2.1.2

Cancel the common factors.

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Step 4.2.1.2.1

Factor $2$ out of $4$ .

$f (4) = \frac{2 (5 \ln ((4) - 1))}{2 (2)}$

Step 4.2.1.2.2

Cancel the common factor.

$f (4) = \frac{2 (5 \ln ((4) - 1))}{2 \cdot 2}$

Step 4.2.1.2.3

Rewrite the expression.

$f (4) = \frac{5 \ln ((4) - 1)}{2}$

Step 4.2.2

Simplify $5 \ln (4 - 1)$ by moving $5$ inside the logarithm.

$f (4) = \frac{\ln ({(4 - 1)}^{5})}{2}$

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Subtract $1$ from $4$ .

$f (4) = \frac{\ln (3^{5})}{2}$

Step 4.2.3.2

Raise $3$ to the power of $5$ .

$f (4) = \frac{\ln (243)}{2}$

Step 4.2.4

Rewrite $\frac{\ln (243)}{2}$ as $\frac{1}{2} \ln (243)$ .

$f (4) = \frac{1}{2} \cdot \ln (243)$

Step 4.2.5

Simplify $\frac{1}{2} \ln (243)$ by moving $\frac{1}{2}$ inside the logarithm.

$f (4) = \ln (243^{\frac{1}{2}})$

Step 4.2.6

The final answer is $\ln (243^{\frac{1}{2}})$ .

$\ln (243^{\frac{1}{2}})$

Step 4.3

Convert $\ln (243^{\frac{1}{2}})$ to decimal.

$y = 2.74653072$

Step 5

The log function can be graphed using the vertical asymptote at $x = 1$ and the points $(2, 0), (3, 2.3104906), (4, 2.74653072)$ .

Vertical Asymptote: $x = 1$

$\begin{array}{c} x & y \\ 2 & 0 \\ 3 & 2.31 \\ 4 & 2.747 \end{array}$

Step 6