Calculus Examples

$\ln (1 - 7 x)$

Step 1

Find the asymptotes.

Tap for more steps...

Step 1.1

Set the argument of the logarithm equal to zero.

$1 - 7 x = 0$

Step 1.2

Solve for $x$ .

Tap for more steps...

Step 1.2.1

Subtract $1$ from both sides of the equation.

$- 7 x = - 1$

Step 1.2.2

Divide each term in $- 7 x = - 1$ by $- 7$ and simplify.

Tap for more steps...

Step 1.2.2.1

Divide each term in $- 7 x = - 1$ by $- 7$ .

$\frac{- 7 x}{- 7} = \frac{- 1}{- 7}$

Step 1.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.2.1

Cancel the common factor of $- 7$ .

Tap for more steps...

Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} = \frac{- 1}{- 7}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 7}$

Step 1.2.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{1}{7}$

Step 1.3

The vertical asymptote occurs at $x = \frac{1}{7}$ .

Vertical Asymptote: $x = \frac{1}{7}$

Step 2

Find the point at $x = 0$ .

Tap for more steps...

Step 2.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \ln (1 - 7 \cdot 0)$

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Multiply $- 7$ by $0$ .

$f (0) = \ln (1 + 0)$

Step 2.2.2

Add $1$ and $0$ .

$f (0) = \ln (1)$

Step 2.2.3

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

$f (0) = 0$

Step 2.2.4

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$= 0$

Step 3

Find the point at $x = - 1$ .

Tap for more steps...

Step 3.1

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

$f (- 1) = \ln (1 - 7 \cdot - 1)$

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Multiply $- 7$ by $- 1$ .

$f (- 1) = \ln (1 + 7)$

Step 3.2.2

Add $1$ and $7$ .

$f (- 1) = \ln (8)$

Step 3.2.3

The final answer is $\ln (8)$ .

$\ln (8)$

Step 3.3

Convert $\ln (8)$ to decimal.

$= 2.07944154$

Step 4

Find the point at $x = - 2$ .

Tap for more steps...

Step 4.1

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

$f (- 2) = \ln (1 - 7 \cdot - 2)$

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Multiply $- 7$ by $- 2$ .

$f (- 2) = \ln (1 + 14)$

Step 4.2.2

Add $1$ and $14$ .

$f (- 2) = \ln (15)$

Step 4.2.3

The final answer is $\ln (15)$ .

$\ln (15)$

Step 4.3

Convert $\ln (15)$ to decimal.

$= 2.7080502$

Step 5

The log function can be graphed using the vertical asymptote at $x = \frac{1}{7}$ and the points $(0, 0), (- 1, 2.07944154), (- 2, 2.7080502)$ .

Vertical Asymptote: $x = \frac{1}{7}$

$\begin{array}{c} x & y \\ - 2 & 2.708 \\ - 1 & 2.079 \\ 0 & 0 \end{array}$

Step 6