Algebra Examples

$y = \log_{4} (- x)$

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.

$f (- 1) = \log_{4} (- (- 1))$

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $- 1$ by $- 1$ .

$f (- 1) = \log_{4} (1)$

Step 2.2.2

Logarithm base $4$ 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.

$f (- 2) = \log_{4} (- (- 2))$

Step 3.2

Simplify the result.

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Step 3.2.1

Multiply $- 1$ by $- 2$ .

$f (- 2) = \log_{4} (2)$

Step 3.2.2

Logarithm base $4$ of $2$ is $\frac{1}{2}$ .

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Step 3.2.2.1

Rewrite as an equation.

$\log_{4} (2) = x$

Step 3.2.2.2

Rewrite $\log_{4} (2) = x$ 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 $b^{y} = x$ .

$4^{x} = 2$

Step 3.2.2.3

Create expressions in the equation that all have equal bases.

${(2^{2})}^{x} = 2^{1}$

Step 3.2.2.4

Rewrite ${(2^{2})}^{x}$ as $2^{2 x}$ .

$2^{2 x} = 2^{1}$

Step 3.2.2.5

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

$2 x = 1$

Step 3.2.2.6

Solve for $x$ .

$x = \frac{1}{2}$

Step 3.2.2.7

The variable $x$ is equal to $\frac{1}{2}$ .

$f (- 2) = \frac{1}{2}$

Step 3.2.3

The final answer is $\frac{1}{2}$ .

$\frac{1}{2}$

Step 3.3

Convert $\frac{1}{2}$ to decimal.

$= 0.5$

Step 4

Find the point at $x = - 3$ .

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Step 4.1

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

$f (- 3) = \log_{4} (- (- 3))$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $- 1$ by $- 3$ .

$f (- 3) = \log_{4} (3)$

Step 4.2.2

The final answer is $\log_{4} (3)$ .

$\log_{4} (3)$

Step 4.3

Convert $\log_{4} (3)$ to decimal.

$= 0.79248125$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(- 1, 0), (- 2, 0.5), (- 3, 0.79248125)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ - 3 & 0.792 \\ - 2 & 0.5 \\ - 1 & 0 \end{array}$

Step 6