Algebra Examples

$f (x) = \log_{9} (x)$

Step 1

Write $f (x) = \log_{9} (x)$ as an equation.

$y = \log_{9} (x)$

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.

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Step 3.1

Remove parentheses.

$y = \log_{9} (1)$

Step 3.2

Logarithm base $9$ of $1$ is $0$ .

$y = 0$

Step 3.3

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.

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Step 4.1

Remove parentheses.

$y = \log_{9} (2)$

Step 4.2

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

$(2, \log_{9} (2))$

Step 5

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

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Step 5.1

Remove parentheses.

$y = \log_{9} (3)$

Step 5.2

Logarithm base $9$ of $3$ is $\frac{1}{2}$ .

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Step 5.2.1

Rewrite as an equation.

$y = \log_{9} (3) = x$

Step 5.2.2

Rewrite $\log_{9} (3) = x$ 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 $b^{y} = x$ .

$y = 9^{x} = 3$

Step 5.2.3

Create expressions in the equation that all have equal bases.

$y = {(3^{2})}^{x} = 3^{1}$

Step 5.2.4

Rewrite ${(3^{2})}^{x}$ as $3^{2 x}$ .

$y = 3^{2 x} = 3^{1}$

Step 5.2.5

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

$y = 2 x = 1$

Step 5.2.6

Solve for $x$ .

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

Step 5.2.7

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

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

Step 5.3

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

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

Step 6

These are three possible solutions to the equation.

$(1, 0), (2, \log_{9} (2)), (3, \frac{1}{2})$

Step 7