Calculus Examples

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

Step 1

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

$\sqrt{x} - 1 > 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $1$ to both sides of the inequality.

$\sqrt{x} > 1$

Step 2.2

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{x}}^{2} > 1^{2}$

Step 2.3

Simplify each side of the inequality.

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Step 2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

${(x^{\frac{1}{2}})}^{2} > 1^{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 2.3.2.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 2.3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{2} \cdot 2} > 1^{2}$

Step 2.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.3.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} > 1^{2}$

Step 2.3.2.1.1.2.2

Rewrite the expression.

$x^{1} > 1^{2}$

Step 2.3.2.1.2

Simplify.

$x > 1^{2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

One to any power is one.

$x > 1$

Step 2.4

Find the domain of $\sqrt{x} - 1$ .

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Step 2.4.1

Set the radicand in $\sqrt{x}$ greater than or equal to $0$ to find where the expression is defined.

$x \geq 0$

Step 2.4.2

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

$[0, \infty)$

Step 2.5

The solution consists of all of the true intervals.

$x > 1$

Step 3

Set the radicand in $\sqrt{x}$ greater than or equal to $0$ to find where the expression is defined.

$x \geq 0$

Step 4

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

Interval Notation:

$(1, \infty)$

Set-Builder Notation:

${x | x > 1}$

Step 5