Finite Math Examples

$\log (\sqrt[7]{x}) - \log (\log_{7} ({(x)}^{5}))$

Step 1

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

$\sqrt[7]{x} \leq 0$

Step 2

Solve for $x$ .

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Step 2.1

To remove the radical on the left side of the inequality, raise both sides of the inequality to the power of $7$ .

${\sqrt[7]{x}}^{7} \leq 0^{7}$

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 $\sqrt[7]{x}$ as $x^{\frac{1}{7}}$ .

${(x^{\frac{1}{7}})}^{7} \leq 0^{7}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify ${(x^{\frac{1}{7}})}^{7}$ .

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Step 2.2.2.1.1

Multiply the exponents in ${(x^{\frac{1}{7}})}^{7}$ .

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Step 2.2.2.1.1.1

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

$x^{\frac{1}{7} \cdot 7} \leq 0^{7}$

Step 2.2.2.1.1.2

Cancel the common factor of $7$ .

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Step 2.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{7} \cdot 7} \leq 0^{7}$

Step 2.2.2.1.1.2.2

Rewrite the expression.

$x^{1} \leq 0^{7}$

Step 2.2.2.1.2

Simplify.

$x \leq 0^{7}$

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 $\log_{7} ({(x)}^{5})$ less than or equal to $0$ to find where the expression is undefined.

${(x)}^{5} \leq 0$

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.

$\sqrt[5]{x^{5}} \leq \sqrt[5]{0}$

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.

$x \leq \sqrt[5]{0}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $\sqrt[5]{0}$ .

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Step 4.2.2.1.1

Rewrite $0$ as $0^{5}$ .

$x \leq \sqrt[5]{0^{5}}$

Step 4.2.2.1.2

Pull terms out from under the radical.

$x \leq 0$

Step 5

Set the argument in $\log (\log_{7} ({(x)}^{5}))$ less than or equal to $0$ to find where the expression is undefined.

$\log_{7} ({(x)}^{5}) \leq 0$

Step 6

Solve for $x$ .

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Step 6.1

Remove parentheses.

$\log_{7} (x^{5}) \leq 0$

Step 6.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = 1$

Step 6.3

Find the domain of $\log_{7} ({(x)}^{5})$ .

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Step 6.3.1

Set the argument in $\log_{7} ({(x)}^{5})$ greater than $0$ to find where the expression is defined.

${(x)}^{5} > 0$

Step 6.3.2

Solve for $x$ .

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Step 6.3.2.1

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

$\sqrt[5]{x^{5}} > \sqrt[5]{0}$

Step 6.3.2.2

Simplify the equation.

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Step 6.3.2.2.1

Simplify the left side.

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Step 6.3.2.2.1.1

Pull terms out from under the radical.

$x > \sqrt[5]{0}$

Step 6.3.2.2.2

Simplify the right side.

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Step 6.3.2.2.2.1

Simplify $\sqrt[5]{0}$ .

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Step 6.3.2.2.2.1.1

Rewrite $0$ as $0^{5}$ .

$x > \sqrt[5]{0^{5}}$

Step 6.3.2.2.2.1.2

Pull terms out from under the radical.

$x > 0$

Step 6.3.3

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

$(0, \infty)$

Step 6.4

Use each root to create test intervals.

$x < 0$

$0 < x < 1$

$x > 1$

Step 6.5

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 6.5.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 6.5.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 6.5.1.2

Replace $x$ with $- 2$ in the original inequality.

$\log_{7} ({(- 2)}^{5}) \leq 0$

Step 6.5.1.3

Determine if the inequality is true.

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Step 6.5.1.3.1

The equation cannot be solved because it is undefined.

$Undefined$

Step 6.5.1.3.2

The left side has no solution, which means that the given statement is false.

False

Step 6.5.2

Test a value on the interval $0 < x < 1$ to see if it makes the inequality true.

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Step 6.5.2.1

Choose a value on the interval $0 < x < 1$ and see if this value makes the original inequality true.

$x = 0.5$

Step 6.5.2.2

Replace $x$ with $0.5$ in the original inequality.

$\log_{7} ({(0.5)}^{5}) \leq 0$

Step 6.5.2.3

The left side $- 1.78103593$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 6.5.3

Test a value on the interval $x > 1$ to see if it makes the inequality true.

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Step 6.5.3.1

Choose a value on the interval $x > 1$ and see if this value makes the original inequality true.

$x = 4$

Step 6.5.3.2

Replace $x$ with $4$ in the original inequality.

$\log_{7} ({(4)}^{5}) \leq 0$

Step 6.5.3.3

The left side $3.56207187$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 6.5.4

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

$x < 0$ False

$0 < x < 1$ True

$x > 1$ False

$x < 0$ False

$0 < x < 1$ True

$x > 1$ False

Step 6.6

The solution consists of all of the true intervals.

$0 < x \leq 1$

Step 7

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 $0$ .

$x \leq 1$

$(- \infty, 1]$

Step 8