Graph f(x)=2* log base 1/4 of x

$f (x) = 2 \cdot \log_{\frac{1}{4}} (x)$

Step 1

Find the asymptotes.

Set the argument of the logarithm equal to zero.

$x^{2} = 0$

Solve for $x$ .

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

The vertical asymptote occurs at $x = 0$ .

Vertical Asymptote: $x = 0$

Step 2

Find the point at $x = 1$ .

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

$f (1) = 2 \cdot \log_{\frac{1}{4}} (1)$

Simplify the result.

Logarithm base $\frac{1}{4}$ of $1$ is $0$ .

$f (1) = 2 \cdot 0$

Multiply $2$ by $0$ .

$f (1) = 0$

The final answer is $0$ .

$0$

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 2$ .

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

$f (2) = 2 \cdot \log_{\frac{1}{4}} (2)$

Simplify the result.

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

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

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

$f (2) = - 1$

The final answer is $- 1$ .

$- 1$

Convert $- 1$ to decimal.

$y = - 1$

Step 4

Find the point at $x = 4$ .

Replace the variable $x$ with $4$ in the expression.

$f (4) = 2 \cdot \log_{\frac{1}{4}} (4)$

Simplify the result.

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

$f (4) = 2 \cdot - 1$

Multiply $2$ by $- 1$ .

$f (4) = - 2$

The final answer is $- 2$ .

$- 2$

Convert $- 2$ to decimal.

$y = - 2$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0), (2, - 1), (4, - 2)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & - 1 \\ 4 & - 2 \end{array}$

Step 6

Graph f(x)=2* log base 1/4 of x

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