Trigonometry Examples

$h x = \ln (- x)$

Step 1

Find the asymptotes.

Tap for more steps...

Step 1.1

Set the argument of the logarithm equal to zero.

$- x = 0$

Step 1.2

Divide each term in $- x = 0$ by $- 1$ and simplify.

Tap for more steps...

Step 1.2.1

Divide each term in $- x = 0$ by $- 1$ .

$\frac{- x}{- 1} = \frac{0}{- 1}$

Step 1.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{0}{- 1}$

Step 1.2.2.2

Divide $x$ by $1$ .

$x = \frac{0}{- 1}$

Step 1.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.1

Divide $0$ by $- 1$ .

$x = 0$

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.

$f (- 1) = \frac{\ln (- (- 1))}{- 1}$

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Move the negative one from the denominator of $\frac{\ln (- (- 1))}{- 1}$ .

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

Step 2.2.2

Rewrite $- 1 \cdot \ln (- (- 1))$ as $- \ln (- (- 1))$ .

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

Step 2.2.3

Multiply $- 1$ by $- 1$ .

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

Step 2.2.4

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

$f (- 1) = - 0$

Step 2.2.5

Multiply $- 1$ by $0$ .

$f (- 1) = 0$

Step 2.2.6

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$= 0$

Step 3

Find the point at $x = - 2$ .

Tap for more steps...

Step 3.1

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

$f (- 2) = \frac{\ln (- (- 2))}{- 2}$

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Rewrite $\frac{\ln (- (- 2))}{- 2}$ as $\frac{1}{- 2} \ln (- (- 2))$ .

$f (- 2) = \frac{1}{- 2} \cdot \ln (- (- 2))$

Step 3.2.2

Simplify $\frac{1}{- 2} \ln (- (- 2))$ by moving $- \frac{1}{- 2}$ inside the logarithm.

$f (- 2) = - \ln ({(- (- 2))}^{- \frac{1}{- 2}})$

Step 3.2.3

Multiply $- 1$ by $- 2$ .

$f (- 2) = - \ln (2^{- \frac{1}{- 2}})$

Step 3.2.4

Move the negative in front of the fraction.

$f (- 2) = - \ln (2^{\frac{1}{2}})$

Step 3.2.5

The final answer is $- \ln (2^{\frac{1}{2}})$ .

$- \ln (2^{\frac{1}{2}})$

Step 3.3

Convert $- \ln (2^{\frac{1}{2}})$ to decimal.

$= - 0.34657359$

Step 4

Find the point at $x = - 3$ .

Tap for more steps...

Step 4.1

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

$f (- 3) = \frac{\ln (- (- 3))}{- 3}$

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Rewrite $\frac{\ln (- (- 3))}{- 3}$ as $\frac{1}{- 3} \ln (- (- 3))$ .

$f (- 3) = \frac{1}{- 3} \cdot \ln (- (- 3))$

Step 4.2.2

Simplify $\frac{1}{- 3} \ln (- (- 3))$ by moving $- \frac{1}{- 3}$ inside the logarithm.

$f (- 3) = - \ln ({(- (- 3))}^{- \frac{1}{- 3}})$

Step 4.2.3

Multiply $- 1$ by $- 3$ .

$f (- 3) = - \ln (3^{- \frac{1}{- 3}})$

Step 4.2.4

Move the negative in front of the fraction.

$f (- 3) = - \ln (3^{\frac{1}{3}})$

Step 4.2.5

The final answer is $- \ln (3^{\frac{1}{3}})$ .

$- \ln (3^{\frac{1}{3}})$

Step 4.3

Convert $- \ln (3^{\frac{1}{3}})$ to decimal.

$= - 0.36620409$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(- 1, 0), (- 2, - 0.34657359), (- 3, - 0.36620409)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ - 3 & - 0.366 \\ - 2 & - 0.347 \\ - 1 & 0 \end{array}$

Step 6