Graph y = log base 4 of 1-x

$y = \log_{4} (1 - x)$

Step 1

Find the asymptotes.

Set the argument of the logarithm equal to zero.

$1 - x = 0$

Solve for $x$ .

Subtract $1$ from both sides of the equation.

$- x = - 1$

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 1$ by $- 1$ .

$x = 1$

The vertical asymptote occurs at $x = 1$ .

Vertical Asymptote: $x = 1$

Step 2

Find the point at $x = 0$ .

Replace the variable $x$ with $0$ in the expression.

$f (0) = \log_{4} (1 - (0))$

Simplify the result.

Subtract $0$ from $1$ .

$f (0) = \log_{4} (1)$

Logarithm base $4$ of $1$ is $0$ .

$f (0) = 0$

The final answer is $0$ .

$0$

Convert $0$ to decimal.

$= 0$

Step 3

Find the point at $x = - 1$ .

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

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

Simplify the result.

Multiply $- 1$ by $- 1$ .

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

Add $1$ and $1$ .

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

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

Rewrite as an equation.

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

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 $b^{y} = x$ .

$4^{x} = 2$

Create expressions in the equation that all have equal bases.

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

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

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

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

$2 x = 1$

Solve for $x$ .

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

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

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

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

$\frac{1}{2}$

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

$= 0.5$

Step 4

Find the point at $x = - 2$ .

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

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

Simplify the result.

Multiply $- 1$ by $- 2$ .

$f (- 2) = \log_{4} (1 + 2)$

Add $1$ and $2$ .

$f (- 2) = \log_{4} (3)$

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

$\log_{4} (3)$

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

$= 0.79248125$

Step 5

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

Vertical Asymptote: $x = 1$

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

Step 6

Graph y = log base 4 of 1-x

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