Find the Domain and Range x=2^y

$x = 2^{y}$

Step 1

Rewrite the equation as $2^{y} = x$ .

$2^{y} = x$

Step 2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (2^{y}) = \ln (x)$

Step 3

Expand $\ln (2^{y})$ by moving $y$ outside the logarithm.

$y \ln (2) = \ln (x)$

Step 4

Divide each term in $y \ln (2) = \ln (x)$ by $\ln (2)$ and simplify.

Divide each term in $y \ln (2) = \ln (x)$ by $\ln (2)$ .

$\frac{y \ln (2)}{\ln (2)} = \frac{\ln (x)}{\ln (2)}$

Simplify the left side.

Cancel the common factor of $\ln (2)$ .

Cancel the common factor.

$\frac{y \ln (2)}{\ln (2)} = \frac{\ln (x)}{\ln (2)}$

Divide $y$ by $1$ .

$y = \frac{\ln (x)}{\ln (2)}$

Step 5

Set the argument in $\ln (x)$ greater than $0$ to find where the expression is defined.

$x > 0$

Step 6

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

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${x | x > 0}$

Step 7

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

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 8

Determine the domain and range.

Domain: $(0, \infty), {x | x > 0}$

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

Step 9