Linear Algebra Examples

$e \ln (8 x^{5}) - \ln (2 x)$

Step 1

Set the argument in $\ln (8 x^{5})$ greater than $0$ to find where the expression is defined.

$8 x^{5} > 0$

Step 2

Solve for $x$ .

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Step 2.1

Divide each term in $8 x^{5} > 0$ by $8$ and simplify.

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Step 2.1.1

Divide each term in $8 x^{5} > 0$ by $8$ .

$\frac{8 x^{5}}{8} > \frac{0}{8}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $8$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{8 x^{5}}{8} > \frac{0}{8}$

Step 2.1.2.1.2

Divide $x^{5}$ by $1$ .

$x^{5} > \frac{0}{8}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $0$ by $8$ .

$x^{5} > 0$

Step 2.2

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 2.3

Simplify the equation.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Pull terms out from under the radical.

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

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

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

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Step 2.3.2.1.1

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

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

Step 2.3.2.1.2

Pull terms out from under the radical.

$x > 0$

Step 3

Set the argument in $\ln (2 x)$ greater than $0$ to find where the expression is defined.

$2 x > 0$

Step 4

Divide each term in $2 x > 0$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 x > 0$ by $2$ .

$\frac{2 x}{2} > \frac{0}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} > \frac{0}{2}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x > \frac{0}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $2$ .

$x > 0$

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