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

Divide each term in $- x = 0$ by $- 1$ and simplify.

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Step 1.2.1

Divide each term in $- x = 0$ by $- 1$ .

$\frac{- x}{- 1} = \frac{0}{- 1}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{0}{- 1}$

Step 1.2.2.2

Divide $x$ by $1$ .

$x = \frac{0}{- 1}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $0$ by $- 1$ .

$x = 0$

Step 1.3

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

Multiply $- 1$ by $- 1$ .

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

Step 2.2.2

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

$f (- 1) = 0$

Step 2.2.3

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$= 0$

Step 3

Find the point at $x = - 2$ .

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Step 3.1

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

$f (- 2) = \log_{3} (- (- 2))$

Step 3.2

Simplify the result.

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Step 3.2.1

Multiply $- 1$ by $- 2$ .

$f (- 2) = \log_{3} (2)$

Step 3.2.2

The final answer is $\log_{3} (2)$ .

$\log_{3} (2)$

Step 3.3

Convert $\log_{3} (2)$ to decimal.

$= 0.63092975$

Step 4

Find the point at $x = - 3$ .

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Step 4.1

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

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

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $- 1$ by $- 3$ .

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

Step 4.2.2

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

$f (- 3) = 1$

Step 4.2.3

The final answer is $1$ .

$1$

Step 4.3

Convert $1$ to decimal.

$= 1$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(- 1, 0), (- 2, 0.63092975), (- 3, 1)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ - 3 & 1 \\ - 2 & 0.631 \\ - 1 & 0 \end{array}$

Step 6