Precalculus Examples

$\ln (2 x)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$2 x = 0$

Step 1.2

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $0$ by $2$ .

$x = 0$

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.

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

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $2$ by $1$ .

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

Step 2.2.2

The final answer is $\ln (2)$ .

$\ln (2)$

Step 2.3

Convert $\ln (2)$ to decimal.

$y = 0.69314718$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

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

Step 3.2

Simplify the result.

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Step 3.2.1

Multiply $2$ by $2$ .

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

Step 3.2.2

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 $x = 3$ .

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Step 4.1

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

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

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $2$ by $3$ .

$f (3) = \ln (6)$

Step 4.2.2

The final answer is $\ln (6)$ .

$\ln (6)$

Step 4.3

Convert $\ln (6)$ to decimal.

$y = 1.79175946$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0.69314718), (2, 1.38629436), (3, 1.79175946)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0.693 \\ 2 & 1.386 \\ 3 & 1.792 \end{array}$

Step 6