Graph y = log base 1/2 of x

$y = \log_{\frac{1}{2}} (x)$

Step 1

Find the asymptotes.

Set the argument of the logarithm equal to zero.

$x = 0$

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.

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

Simplify the result.

Logarithm base $\frac{1}{2}$ of $1$ is $0$ .

$f (1) = 0$

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.

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

Simplify the result.

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

Rewrite as an equation.

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

Rewrite $\log_{\frac{1}{2}} (2) = x$ 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 $b^{y} = x$ .

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

Create expressions in the equation that all have equal bases.

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

Rewrite ${(2^{- 1})}^{x}$ as $2^{- x}$ .

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

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

$- x = 1$

Solve for $x$ .

$x = - 1$

The variable $x$ is equal to $- 1$ .

$f (2) = - 1$

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.

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

Simplify the result.

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

Rewrite as an equation.

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

Rewrite $\log_{\frac{1}{2}} (4) = x$ 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 $b^{y} = x$ .

${(\frac{1}{2})}^{x} = 4$

Create expressions in the equation that all have equal bases.

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

Rewrite ${(2^{- 1})}^{x}$ as $2^{- x}$ .

$2^{- x} = 2^{2}$

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

$- x = 2$

Solve for $x$ .

$x = - 2$

The variable $x$ is equal to $- 2$ .

$f (4) = - 2$

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 y = log base 1/2 of x

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