Trigonometry Examples

$f (x) = \log (x) - 6$

Step 1

Find the asymptotes.

Tap for more steps...

Step 1.1

Find where the expression $\log (x) - 6$ is undefined.

$x \leq 0$

Step 1.2

Since $\log (x) - 6$ $\to$ $\infty$ as $x$ $\to$ $0$ from the left and $\log (x) - 6$ $\to$ $- \infty$ as $x$ $\to$ $0$ from the right, then $x = 0$ is a vertical asymptote.

$x = 0$

Step 1.3

Ignoring the logarithm, consider the rational function $R (x) = \frac{a x^{n}}{b x^{m}}$ where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

1. If $n < m$ , then the x-axis, $y = 0$ , is the horizontal asymptote.

2. If $n = m$ , then the horizontal asymptote is the line $y = \frac{a}{b}$ .

3. If $n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 1.4

There are no horizontal asymptotes because $Q (x)$ is $1$ .

No Horizontal Asymptotes

Step 1.5

No oblique asymptotes are present for logarithmic and trigonometric functions.

No Oblique Asymptotes

Step 1.6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

No Horizontal Asymptotes

Vertical Asymptotes: $x = 0$

No Horizontal Asymptotes

Step 2

Find the point at $x = 1$ .

Tap for more steps...

Step 2.1

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

$f (1) = \log (1) - 6$

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Logarithm base $10$ of $1$ is $0$ .

$f (1) = 0 - 6$

Step 2.2.2

Subtract $6$ from $0$ .

$f (1) = - 6$

Step 2.2.3

The final answer is $- 6$ .

$- 6$

Step 2.3

Convert $- 6$ to decimal.

$y = - 6$

Step 3

Find the point at $x = 10$ .

Tap for more steps...

Step 3.1

Replace the variable $x$ with $10$ in the expression.

$f (10) = \log (10) - 6$

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Logarithm base $10$ of $10$ is $1$ .

$f (10) = 1 - 6$

Step 3.2.2

Subtract $6$ from $1$ .

$f (10) = - 5$

Step 3.2.3

The final answer is $- 5$ .

$- 5$

Step 3.3

Convert $- 5$ to decimal.

$y = - 5$

Step 4

Find the point at $x = 2$ .

Tap for more steps...

Step 4.1

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

$f (2) = \log (2) - 6$

Step 4.2

The final answer is $\log (2) - 6$ .

$\log (2) - 6$

Step 4.3

Convert $\log (2) - 6$ to decimal.

$y = - 5.69897$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, - 6), (10, - 5), (2, - 5.69897)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & - 6 \\ 2 & - 5.699 \\ 10 & - 5 \end{array}$

Step 6