Find Three Ordered Pair Solutions f(x) = log base 1/2 of x

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

Step 1

Write $f (x) = \log_{\frac{1}{2}} (x)$ as an equation.

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

Step 2

Choose any value for $x$ that is in the domain to plug into the equation.

Step 3

Choose $1$ to substitute in for $x$ to find the ordered pair.

Remove parentheses.

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

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

$y = 0$

Use the $x$ and $y$ values to form the ordered pair.

$(1, 0)$

Step 4

Choose $2$ to substitute in for $x$ to find the ordered pair.

Remove parentheses.

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

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

Rewrite as an equation.

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

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 $b^{y} = x$ .

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

Create expressions in the equation that all have equal bases.

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

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

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

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

$y = - x = 1$

Solve for $x$ .

$y = x = - 1$

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

$y = - 1$

Use the $x$ and $y$ values to form the ordered pair.

$(2, - 1)$

Step 5

Choose $3$ to substitute in for $x$ to find the ordered pair.

Remove parentheses.

$y = \log_{\frac{1}{2}} (3)$

Use the $x$ and $y$ values to form the ordered pair.

$(3, \log_{\frac{1}{2}} (3))$

Step 6

These are three possible solutions to the equation.

$(1, 0), (2, - 1), (3, \log_{\frac{1}{2}} (3))$

Step 7

Find Three Ordered Pair Solutions f(x) = log base 1/2 of x

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