Calculus Examples

$y = \log (2 x + 7) - 5$

Step 1

Find the asymptotes.

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Step 1.1

Find where the expression $\log (2 x + 7) - 5$ is undefined.

$x \leq - \frac{7}{2}$

Step 1.2

Since $\log (2 x + 7) - 5$ $\to$ $\infty$ as $x$ $\to$ $- \frac{7}{2}$ from the left and $\log (2 x + 7) - 5$ $\to$ $- \infty$ as $x$ $\to$ $- \frac{7}{2}$ from the right, then $x = - \frac{7}{2}$ is a vertical asymptote.

$x = - \frac{7}{2}$

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: $x = - \frac{7}{2}$

No Horizontal Asymptotes

Vertical Asymptotes: $x = - \frac{7}{2}$

No Horizontal Asymptotes

Step 2

Find the point at $x = - 2$ .

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Step 2.1

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

$f (- 2) = \log (2 (- 2) + 7) - 5$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $2$ by $- 2$ .

$f (- 2) = \log (- 4 + 7) - 5$

Step 2.2.1.2

Add $- 4$ and $7$ .

$f (- 2) = \log (3) - 5$

Step 2.2.2

The final answer is $\log (3) - 5$ .

$\log (3) - 5$

Step 2.3

Convert $\log (3) - 5$ to decimal.

$y = - 4.52287874$

Step 3

Find the point at $x = - 1$ .

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Step 3.1

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

$f (- 1) = \log (2 (- 1) + 7) - 5$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $2$ by $- 1$ .

$f (- 1) = \log (- 2 + 7) - 5$

Step 3.2.1.2

Add $- 2$ and $7$ .

$f (- 1) = \log (5) - 5$

Step 3.2.2

The final answer is $\log (5) - 5$ .

$\log (5) - 5$

Step 3.3

Convert $\log (5) - 5$ to decimal.

$y = - 4.30102999$

Step 4

Find the point at $x = 0$ .

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Step 4.1

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

$f (0) = \log (2 (0) + 7) - 5$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $2$ by $0$ .

$f (0) = \log (0 + 7) - 5$

Step 4.2.1.2

Add $0$ and $7$ .

$f (0) = \log (7) - 5$

Step 4.2.2

The final answer is $\log (7) - 5$ .

$\log (7) - 5$

Step 4.3

Convert $\log (7) - 5$ to decimal.

$y = - 4.15490195$

Step 5

The log function can be graphed using the vertical asymptote at $x = - \frac{7}{2}$ and the points $(- 2, - 4.52287874), (- 1, - 4.30102999), (0, - 4.15490195)$ .

Vertical Asymptote: $x = - \frac{7}{2}$

$\begin{array}{c} x & y \\ - 2 & - 4.523 \\ - 1 & - 4.301 \\ 0 & - 4.155 \end{array}$

Step 6