Algebra Examples

$f (x) = \log_{2} (x + 3)$

Step 1

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

$y = \log_{2} ((- 2) + 3)$

Step 2

Solve the equation for $y$ .

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Step 2.1

Remove parentheses.

$y = \log_{2} (- 2 + 3)$

Step 2.2

Remove parentheses.

$y = \log_{2} ((- 2) + 3)$

Step 2.3

Simplify $\log_{2} ((- 2) + 3)$ .

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Step 2.3.1

Add $- 2$ and $3$ .

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

Step 2.3.2

Logarithm base $2$ of $1$ is $0$ .

$y = 0$

Step 3

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

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

Step 4

Solve the equation for $y$ .

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Step 4.1

Remove parentheses.

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

Step 4.2

Remove parentheses.

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

Step 4.3

Simplify $\log_{2} ((- 1) + 3)$ .

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Step 4.3.1

Add $- 1$ and $3$ .

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

Step 4.3.2

Logarithm base $2$ of $2$ is $1$ .

$y = 1$

Step 5

Substitute $0$ for $x$ and find the result for $y$ .

$y = \log_{2} ((0) + 3)$

Step 6

Solve the equation for $y$ .

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Step 6.1

Remove parentheses.

$y = \log_{2} (0 + 3)$

Step 6.2

Remove parentheses.

$y = \log_{2} ((0) + 3)$

Step 6.3

Add $0$ and $3$ .

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

Step 7

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

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

Step 8

Solve the equation for $y$ .

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Step 8.1

Remove parentheses.

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

Step 8.2

Remove parentheses.

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

Step 8.3

Simplify $\log_{2} ((1) + 3)$ .

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Step 8.3.1

Add $1$ and $3$ .

$y = \log_{2} (4)$

Step 8.3.2

Logarithm base $2$ of $4$ is $2$ .

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Step 8.3.2.1

Rewrite as an equation.

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

Step 8.3.2.2

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

$y = 2^{x} = 4$

Step 8.3.2.3

Create equivalent expressions in the equation that all have equal bases.

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

Step 8.3.2.4

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

$y = x = 2$

Step 8.3.2.5

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

$y = 2$

Step 9

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

$y = \log_{2} ((2) + 3)$

Step 10

Solve the equation for $y$ .

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Step 10.1

Remove parentheses.

$y = \log_{2} (2 + 3)$

Step 10.2

Remove parentheses.

$y = \log_{2} ((2) + 3)$

Step 10.3

Add $2$ and $3$ .

$y = \log_{2} (5)$

Step 11

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

$\begin{array}{c} x & y \\ - 2 & 0 \\ - 1 & 1 \\ 0 & \log_{2} (3) \\ 1 & 2 \\ 2 & \log_{2} (5) \end{array}$

Step 12