Algebra Examples

$f (x) = \log_{3} (\frac{x}{3})$

Step 1

Find the asymptotes.

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Step 1.1

Find where the expression $\log_{3} (x) - 1$ is undefined.

$x \leq 0$

Step 1.2

Since $\log_{3} (x) - 1$ $\to$ $\infty$ as $x$ $\to$ $0$ from the left and $\log_{3} (x) - 1$ $\to$ $- \infty$ 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.

$f (1) = \log_{3} (\frac{1}{3})$

Step 2.2

Simplify the result.

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Step 2.2.1

Logarithm base $3$ of $\frac{1}{3}$ is $- 1$ .

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Step 2.2.1.1

Rewrite as an equation.

$\log_{3} (\frac{1}{3}) = x$

Step 2.2.1.2

Rewrite $\log_{3} (\frac{1}{3}) = x$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ does not equal $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$3^{x} = \frac{1}{3}$

Step 2.2.1.3

Create equivalent expressions in the equation that all have equal bases.

$3^{x} = 3^{- 1}$

Step 2.2.1.4

Since the bases are the same, the two expressions are only equal if the exponents are also equal.

$x = - 1$

Step 2.2.1.5

The variable $x$ is equal to $- 1$ .

$f (1) = - 1$

Step 2.2.2

The final answer is $- 1$ .

$- 1$

Step 2.3

Convert $- 1$ to decimal.

$y = - 1$

Step 3

Find the point at $x = 3$ .

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Step 3.1

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

$f (3) = \log_{3} (\frac{3}{3})$

Step 3.2

Simplify the result.

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Step 3.2.1

Divide $3$ by $3$ .

$f (3) = \log_{3} (1)$

Step 3.2.2

Logarithm base $3$ of $1$ is $0$ .

$f (3) = 0$

Step 3.2.3

The final answer is $0$ .

$0$

Step 3.3

Convert $0$ to decimal.

$y = 0$

Step 4

Find the point at $x = 9$ .

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Step 4.1

Replace the variable $x$ with $9$ in the expression.

$f (9) = \log_{3} (\frac{9}{3})$

Step 4.2

Simplify the result.

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Step 4.2.1

Logarithm base $3$ of $\frac{9}{3}$ is $1$ .

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Step 4.2.1.1

Rewrite as an equation.

$\log_{3} (\frac{9}{3}) = x$

Step 4.2.1.2

Rewrite $\log_{3} (\frac{9}{3}) = x$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ does not equal $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$3^{x} = \frac{9}{3}$

Step 4.2.1.3

Create equivalent expressions in the equation that all have equal bases.

$3^{x} = 3^{1}$

Step 4.2.1.4

Since the bases are the same, the two expressions are only equal if the exponents are also equal.

$x = 1$

Step 4.2.1.5

The variable $x$ is equal to $1$ .

$f (9) = 1$

Step 4.2.2

The final answer is $1$ .

$1$

Step 4.3

Convert $1$ to decimal.

$y = 1$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, - 1), (3, 0), (9, 1)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & - 1 \\ 3 & 0 \\ 9 & 1 \end{array}$

Step 6