Calculus Examples

$y = 2 \ln (x - 2)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

${(x - 2)}^{2} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 1.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.3

The vertical asymptote occurs at $x = 2$ .

Vertical Asymptote: $x = 2$

Step 2

Find the point at $x = 3$ .

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Step 2.1

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

$f (3) = 2 \ln ((3) - 2)$

Step 2.2

Simplify the result.

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Step 2.2.1

Subtract $2$ from $3$ .

$f (3) = 2 \ln (1)$

Step 2.2.2

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

$f (3) = 2 \cdot 0$

Step 2.2.3

Multiply $2$ by $0$ .

$f (3) = 0$

Step 2.2.4

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 4$ .

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Step 3.1

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

$f (4) = 2 \ln ((4) - 2)$

Step 3.2

Simplify the result.

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Step 3.2.1

Subtract $2$ from $4$ .

$f (4) = 2 \ln (2)$

Step 3.2.2

Simplify $2 \ln (2)$ by moving $2$ inside the logarithm.

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

Step 3.2.3

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

$f (4) = \ln (4)$

Step 3.2.4

The final answer is $\ln (4)$ .

$\ln (4)$

Step 3.3

Convert $\ln (4)$ to decimal.

$y = 1.38629436$

Step 4

Find the point at $x = 5$ .

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Step 4.1

Replace the variable $x$ with $5$ in the expression.

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

Step 4.2

Simplify the result.

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Step 4.2.1

Subtract $2$ from $5$ .

$f (5) = 2 \ln (3)$

Step 4.2.2

Simplify $2 \ln (3)$ by moving $2$ inside the logarithm.

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

Step 4.2.3

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

$f (5) = \ln (9)$

Step 4.2.4

The final answer is $\ln (9)$ .

$\ln (9)$

Step 4.3

Convert $\ln (9)$ to decimal.

$y = 2.19722457$

Step 5

The log function can be graphed using the vertical asymptote at $x = 2$ and the points $(3, 0), (4, 1.38629436), (5, 2.19722457)$ .

Vertical Asymptote: $x = 2$

$\begin{array}{c} x & y \\ 3 & 0 \\ 4 & 1.386 \\ 5 & 2.197 \end{array}$

Step 6