Calculus Examples

$\sqrt{\ln (x y - 1)}$

Step 1

Set the argument in $\ln (x y - 1)$ greater than $0$ to find where the expression is defined.

$x y - 1 > 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $1$ to both sides of the inequality.

$x y > 1$

Step 2.2

Divide each term in $x y > 1$ by $y$ and simplify.

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Step 2.2.1

Divide each term in $x y > 1$ by $y$ .

$\frac{x y}{y} > \frac{1}{y}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $y$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{x y}{y} > \frac{1}{y}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{1}{y}$

Step 2.3

Use each root to create test intervals.

$x < \frac{1}{y}$

$x > \frac{1}{y}$

Step 2.4

Compare the intervals to determine which ones satisfy the original inequality.

Step 2.5

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution

Step 3

Set the radicand in $\sqrt{\ln (x y - 1)}$ greater than or equal to $0$ to find where the expression is defined.

$\ln (x y - 1) \geq 0$

Step 4

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 5