Algebra Examples

$\log_{2} (x + 4) - 3$

Step 1

Write $\log_{2} (x + 4) - 3$ as an equation.

$y = \log_{2} (x + 4) - 3$

Step 2

Find the x-intercepts.

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Step 2.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \log_{2} (x + 4) - 3$

Step 2.2

Solve the equation.

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Step 2.2.1

Rewrite the equation as $\log_{2} (x + 4) - 3 = 0$ .

$\log_{2} (x + 4) - 3 = 0$

Step 2.2.2

Add $3$ to both sides of the equation.

$\log_{2} (x + 4) = 3$

Step 2.2.3

Rewrite $\log_{2} (x + 4) = 3$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$2^{3} = x + 4$

Step 2.2.4

Solve for $x$ .

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Step 2.2.4.1

Rewrite the equation as $x + 4 = 2^{3}$ .

$x + 4 = 2^{3}$

Step 2.2.4.2

Raise $2$ to the power of $3$ .

$x + 4 = 8$

Step 2.2.4.3

Move all terms not containing $x$ to the right side of the equation.

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Step 2.2.4.3.1

Subtract $4$ from both sides of the equation.

$x = 8 - 4$

Step 2.2.4.3.2

Subtract $4$ from $8$ .

$x = 4$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(4, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \log_{2} ((0) + 4) - 3$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = \log_{2} (0 + 4) - 3$

Step 3.2.2

Remove parentheses.

$y = \log_{2} ((0) + 4) - 3$

Step 3.2.3

Simplify $\log_{2} ((0) + 4) - 3$ .

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Add $0$ and $4$ .

$y = \log_{2} (4) - 3$

Step 3.2.3.1.2

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

$y = 2 - 3$

Step 3.2.3.2

Subtract $3$ from $2$ .

$y = - 1$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 1)$

Step 4

List the intersections.

x-intercept(s): $(4, 0)$

y-intercept(s): $(0, - 1)$

Step 5