Algebra Examples

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

Step 1

Find the x-intercepts.

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Step 1.1

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

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

Step 1.2.2

Rewrite $\log_{2} (2 x + 3) = 0$ 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 $b^{y} = x$ .

$2^{0} = 2 x + 3$

Step 1.2.3

Solve for $x$ .

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Step 1.2.3.1

Rewrite the equation as $2 x + 3 = 2^{0}$ .

$2 x + 3 = 2^{0}$

Step 1.2.3.2

Anything raised to $0$ is $1$ .

$2 x + 3 = 1$

Step 1.2.3.3

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

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Step 1.2.3.3.1

Subtract $3$ from both sides of the equation.

$2 x = 1 - 3$

Step 1.2.3.3.2

Subtract $3$ from $1$ .

$2 x = - 2$

Step 1.2.3.4

Divide each term in $2 x = - 2$ by $2$ and simplify.

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Step 1.2.3.4.1

Divide each term in $2 x = - 2$ by $2$ .

$\frac{2 x}{2} = \frac{- 2}{2}$

Step 1.2.3.4.2

Simplify the left side.

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Step 1.2.3.4.2.1

Cancel the common factor of $2$ .

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Step 1.2.3.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 2}{2}$

Step 1.2.3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{2}$

Step 1.2.3.4.3

Simplify the right side.

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Step 1.2.3.4.3.1

Divide $- 2$ by $2$ .

$x = - 1$

Step 1.3

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

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Step 2.1

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

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

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

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Step 2.2.2.1

Multiply $2$ by $0$ .

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

Step 2.2.2.2

Add $0$ and $3$ .

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

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \log_{2} (3))$

Step 3

List the intersections.

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

y-intercept(s): $(0, \log_{2} (3))$

Step 4