Pre-Algebra Examples

$\ln (e^{x} - 3)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$e^{x} - 3 = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Add $3$ to both sides of the equation.

$e^{x} = 3$

Step 1.2.2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{x}) = \ln (3)$

Step 1.2.3

Expand the left side.

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Step 1.2.3.1

Expand $\ln (e^{x})$ by moving $x$ outside the logarithm.

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

Step 1.2.3.2

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

$x \cdot 1 = \ln (3)$

Step 1.2.3.3

Multiply $x$ by $1$ .

$x = \ln (3)$

Step 1.3

The vertical asymptote occurs at $x = \ln (3)$ .

Vertical Asymptote: $x = \ln (3)$

Step 2

Find the point at $x = 3$ .

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Step 2.1

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

$f (3) = \ln (e^{3} - 3)$

Step 2.2

The final answer is $\ln (e^{3} - 3)$ .

$\ln (e^{3} - 3)$

Step 2.3

Convert $\ln (e^{3} - 3)$ to decimal.

$y = 2.83823231$

Step 3

Find the point at $x = 4$ .

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Step 3.1

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

$f (4) = \ln (e^{4} - 3)$

Step 3.2

The final answer is $\ln (e^{4} - 3)$ .

$\ln (e^{4} - 3)$

Step 3.3

Convert $\ln (e^{4} - 3)$ to decimal.

$y = 3.94348581$

Step 4

Find the point at $x = 5$ .

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Step 4.1

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

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

Step 4.2

The final answer is $\ln (e^{5} - 3)$ .

$\ln (e^{5} - 3)$

Step 4.3

Convert $\ln (e^{5} - 3)$ to decimal.

$y = 4.97957906$

Step 5

The log function can be graphed using the vertical asymptote at $x = \ln (3)$ and the points $(3, 2.83823231), (4, 3.94348581), (5, 4.97957906)$ .

Vertical Asymptote: $x = \ln (3)$

$\begin{array}{c} x & y \\ 3 & 2.838 \\ 4 & 3.943 \\ 5 & 4.98 \end{array}$

Step 6