Pre-Algebra Examples

$\ln (e x)$

Step 1

Find the asymptotes.

Tap for more steps...

Step 1.1

Set the argument of the logarithm equal to zero.

$e x = 0$

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.1

Cancel the common factor of $e$ .

Tap for more steps...

Step 1.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

Step 1.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.1

Divide $0$ by $e$ .

$x = 0$

Step 1.3

The vertical asymptote occurs at $x = 0$ .

Vertical Asymptote: $x = 0$

Step 2

Find the point at $x = 1$ .

Tap for more steps...

Step 2.1

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

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

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Multiply $e$ by $1$ .

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

Step 2.2.2

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

$f (1) = 1$

Step 2.2.3

The final answer is $1$ .

$1$

Step 2.3

Convert $1$ to decimal.

$y = 1$

Step 3

Find the point at $x = 2$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Move $2$ to the left of $e$ .

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

Step 3.2.2

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

$\ln (2 e)$

Step 3.3

Convert $\ln (2 e)$ to decimal.

$y = 1.69314718$

Step 4

Find the point at $x = 3$ .

Tap for more steps...

Step 4.1

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

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

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Move $3$ to the left of $e$ .

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

Step 4.2.2

The final answer is $\ln (3 e)$ .

$\ln (3 e)$

Step 4.3

Convert $\ln (3 e)$ to decimal.

$y = 2.09861228$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 1), (2, 1.69314718), (3, 2.09861228)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 1 \\ 2 & 1.693 \\ 3 & 2.099 \end{array}$

Step 6