Algebra Examples

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

Step 1

Substitute $1$ for $x$ and find the result for $y$ .

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

Step 2

Solve the equation for $y$ .

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Step 2.1

Remove parentheses.

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

Step 2.2

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

$y = 0$

Step 3

Substitute $2$ for $x$ and find the result for $y$ .

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

Step 4

Solve the equation for $y$ .

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Step 4.1

Remove parentheses.

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

Step 4.2

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

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Step 4.2.1

Rewrite as an equation.

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

Step 4.2.2

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 $b^{y} = x$ .

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

Step 4.2.3

Create expressions in the equation that all have equal bases.

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

Step 4.2.4

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

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

Step 4.2.5

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

$y = - x = 1$

Step 4.2.6

Solve for $x$ .

$y = x = - 1$

Step 4.2.7

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

$y = - 1$

Step 5

This is a table of possible values to use when graphing the equation.

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

Step 6