Finite Math Examples

$\log_{7} (\sqrt{x}) - \log_{7} (x^{3})$

Step 1

Set the argument in $\log_{7} (\sqrt{x})$ less than or equal to $0$ to find where the expression is undefined.

$\sqrt{x} \leq 0$

Step 2

Solve for $x$ .

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Step 2.1

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

${\sqrt{x}}^{2} \leq 0^{2}$

Step 2.2

Simplify each side of the inequality.

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Step 2.2.1

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

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

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

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Step 2.2.2.1.1

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

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Step 2.2.2.1.1.1

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

$x^{\frac{1}{2} \cdot 2} \leq 0^{2}$

Step 2.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} \leq 0^{2}$

Step 2.2.2.1.1.2.2

Rewrite the expression.

$x^{1} \leq 0^{2}$

Step 2.2.2.1.2

Simplify.

$x \leq 0^{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Raising $0$ to any positive power yields $0$ .

$x \leq 0$

Step 3

Set the argument in $\log_{7} (x^{3})$ less than or equal to $0$ to find where the expression is undefined.

$x^{3} \leq 0$

Step 4

Solve for $x$ .

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Step 4.1

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt[3]{x^{3}} \leq \sqrt[3]{0}$

Step 4.2

Simplify the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Pull terms out from under the radical.

$x \leq \sqrt[3]{0}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 4.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$x \leq \sqrt[3]{0^{3}}$

Step 4.2.2.1.2

Pull terms out from under the radical.

$x \leq 0$

Step 5

Set the radicand in $\sqrt{x}$ less than $0$ to find where the expression is undefined.

$x < 0$

Step 6

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq 0$

$(- \infty, 0]$

Step 7