Find Three Ordered Pair Solutions f(x)=4 log base 1/2 of x

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

Step 1

Write $f (x) = 4 \log_{\frac{1}{2}} (x)$ as an equation.

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

Step 2

Simplify $4 \log_{\frac{1}{2}} (x)$ by moving $4$ inside the logarithm.

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

Step 3

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

Step 4

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

Remove parentheses.

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

Remove parentheses.

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

Simplify $\log_{\frac{1}{2}} ({(1)}^{4})$ .

One to any power is one.

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

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 5

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

Remove parentheses.

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

Remove parentheses.

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

Simplify $\log_{\frac{1}{2}} ({(2)}^{4})$ .

Raise $2$ to the power of $4$ .

$y = \log_{\frac{1}{2}} (16)$

Logarithm base $\frac{1}{2}$ of $16$ is $- 4$ .

Rewrite as an equation.

$y = \log_{\frac{1}{2}} (16) = x$

Rewrite $\log_{\frac{1}{2}} (16) = x$ 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 $b^{y} = x$ .

$y = {(\frac{1}{2})}^{x} = 16$

Create expressions in the equation that all have equal bases.

$y = {(2^{- 1})}^{x} = 2^{4}$

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

$y = 2^{- x} = 2^{4}$

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

$y = - x = 4$

Solve for $x$ .

$y = x = - 4$

The variable $x$ is equal to $- 4$ .

$y = - 4$

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

$(2, - 4)$

Step 6

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

Remove parentheses.

$y = \log_{\frac{1}{2}} (3^{4})$

Remove parentheses.

$y = \log_{\frac{1}{2}} ({(3)}^{4})$

Raise $3$ to the power of $4$ .

$y = \log_{\frac{1}{2}} (81)$

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

$(3, \log_{\frac{1}{2}} (81))$

Step 7

These are three possible solutions to the equation.

$(1, 0), (2, - 4), (3, \log_{\frac{1}{2}} (81))$

Step 8

Find Three Ordered Pair Solutions f(x)=4 log base 1/2 of x

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