Precalculus Examples

$f (x) = 2 + \log_{5} (x)$

Step 1

Find the asymptotes.

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Step 1.1

Find where the expression $2 + \log_{5} (x)$ is undefined.

$x \leq 0$

Step 1.2

Since $2 + \log_{5} (x)$ $\to$ $\infty$ as $x$ $\to$ $0$ from the left and $2 + \log_{5} (x)$ $\to$ $- \infty$ 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 $x = 1$ .

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Step 2.1

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

$f (1) = 2 + \log_{5} (1)$

Step 2.2

Simplify the result.

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Step 2.2.1

Logarithm base $5$ of $1$ is $0$ .

$f (1) = 2 + 0$

Step 2.2.2

Add $2$ and $0$ .

$f (1) = 2$

Step 2.2.3

The final answer is $2$ .

$2$

Step 2.3

Convert $2$ to decimal.

$y = 2$

Step 3

Find the point at $x = 5$ .

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Step 3.1

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

$f (5) = 2 + \log_{5} (5)$

Step 3.2

Simplify the result.

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Step 3.2.1

Logarithm base $5$ of $5$ is $1$ .

$f (5) = 2 + 1$

Step 3.2.2

Add $2$ and $1$ .

$f (5) = 3$

Step 3.2.3

The final answer is $3$ .

$3$

Step 3.3

Convert $3$ to decimal.

$y = 3$

Step 4

Find the point at $x = 2$ .

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Step 4.1

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

$f (2) = 2 + \log_{5} (2)$

Step 4.2

The final answer is $2 + \log_{5} (2)$ .

$2 + \log_{5} (2)$

Step 4.3

Convert $2 + \log_{5} (2)$ to decimal.

$y = 2.43067655$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 2), (5, 3), (2, 2.43067655)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 2 \\ 2 & 2.431 \\ 5 & 3 \end{array}$

Step 6