Find the Domain and Range y = log base 2 of 2x+3

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

Set the argument in $\log_{2} (2 x + 3)$ greater than $0$ to find where the expression is defined.

$2 x + 3 > 0$

Solve for $x$ .

Subtract $3$ from both sides of the inequality.

$2 x > - 3$

Divide each term in $2 x > - 3$ by $2$ and simplify.

Divide each term in $2 x > - 3$ by $2$ .

$\frac{2 x}{2} > \frac{- 3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} > \frac{- 3}{2}$

Divide $x$ by $1$ .

$x > \frac{- 3}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x > - \frac{3}{2}$

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \frac{3}{2}, \infty)$

Set-Builder Notation:

${x | x > - \frac{3}{2}}$

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Determine the domain and range.

Domain: $(- \frac{3}{2}, \infty), {x | x > - \frac{3}{2}}$

Range: $(- \infty, \infty), {y | y \in ℝ}$