Finite Math Examples

$\log_{5} (3 x^{\frac{1}{2}})$

Step 1

Convert expressions with fractional exponents to radicals.

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Step 1.1

Apply the rule

$x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\log_{5} (3 \sqrt{x^{1}})$

Step 1.2

Anything raised to

$1$ is the base itself.

$\log_{5} (3 \sqrt{x})$

Step 2

Set the argument in

$\log_{5} (3 \sqrt{x})$ greater than

$0$ to find where the expression is defined.

$3 \sqrt{x} > 0$

Step 3

Solve for

$x$ .

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Step 3.1

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

${(3 \sqrt{x})}^{2} > 0^{2}$

Step 3.2

Simplify each side of the inequality.

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Step 3.2.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{x}$ as

$x^{\frac{1}{2}}$ .

${(3 x^{\frac{1}{2}})}^{2} > 0^{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify

${(3 x^{\frac{1}{2}})}^{2}$ .

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Step 3.2.2.1.1

Apply the product rule to

$3 x^{\frac{1}{2}}$ .

$3^{2} {(x^{\frac{1}{2}})}^{2} > 0^{2}$

Step 3.2.2.1.2

Raise

$3$ to the power of

$2$ .

$9 {(x^{\frac{1}{2}})}^{2} > 0^{2}$

Step 3.2.2.1.3

Multiply the exponents in

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

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Step 3.2.2.1.3.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$9 x^{\frac{1}{2} \cdot 2} > 0^{2}$

Step 3.2.2.1.3.2

Cancel the common factor of

$2$ .

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Step 3.2.2.1.3.2.1

Cancel the common factor.

$9 x^{\frac{1}{2} \cdot 2} > 0^{2}$

Step 3.2.2.1.3.2.2

Rewrite the expression.

$9 x^{1} > 0^{2}$

Step 3.2.2.1.4

Simplify.

$9 x > 0^{2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Raising

$0$ to any positive power yields

$0$ .

$9 x > 0$

Step 3.3

Divide each term in

$9 x > 0$ by

$9$ and simplify.

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Step 3.3.1

Divide each term in

$9 x > 0$ by

$9$ .

$\frac{9 x}{9} > \frac{0}{9}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of

$9$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{9 x}{9} > \frac{0}{9}$

Step 3.3.2.1.2

Divide

$x$ by

$1$ .

$x > \frac{0}{9}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide

$0$ by

$9$ .

$x > 0$

Step 3.4

Find the domain of

$3 \sqrt{x}$ .

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Step 3.4.1

Set the radicand in

$\sqrt{x}$ greater than or equal to

$0$ to find where the expression is defined.

$x \geq 0$

Step 3.4.2

The domain is all values of

$x$ that make the expression defined.

$[0, \infty)$

Step 3.5

The solution consists of all of the true intervals.

$x > 0$

Step 4

Set the radicand in

$\sqrt{x}$ greater than or equal to

$0$ to find where the expression is defined.

$x \geq 0$

Step 5

The domain is all values of

$x$ that make the expression defined.

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${x | x > 0}$

Step 6