Algebra Examples

$y = - \log_{3} (x)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$x = 0$

Step 1.2

The vertical asymptote occurs at $x = 0$ .

Vertical Asymptote: $x = 0$

Step 2

Find the point at $x = 1$ .

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Step 2.1

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

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

Step 2.2

Simplify the result.

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Step 2.2.1

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

$f (1) = - 0$

Step 2.2.2

Multiply $- 1$ by $0$ .

$f (1) = 0$

Step 2.2.3

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 3$ .

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Step 3.1

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

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

Step 3.2

Simplify the result.

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Step 3.2.1

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

$f (3) = - 1 \cdot 1$

Step 3.2.2

Multiply $- 1$ by $1$ .

$f (3) = - 1$

Step 3.2.3

The final answer is $- 1$ .

$- 1$

Step 3.3

Convert $- 1$ to decimal.

$y = - 1$

Step 4

Find the point at $x = 9$ .

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Step 4.1

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

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

Step 4.2

Simplify the result.

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Step 4.2.1

Logarithm base $3$ of $9$ is $2$ .

$f (9) = - 1 \cdot 2$

Step 4.2.2

Multiply $- 1$ by $2$ .

$f (9) = - 2$

Step 4.2.3

The final answer is $- 2$ .

$- 2$

Step 4.3

Convert $- 2$ to decimal.

$y = - 2$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0), (3, - 1), (9, - 2)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0 \\ 3 & - 1 \\ 9 & - 2 \end{array}$

Step 6