Finite Math Examples

$4 \log (u) - 5 \log ({(u)}^{6})$

Step 1

Set the argument in $\log (u)$ less than or equal to $0$ to find where the expression is undefined.

$u \leq 0$

Step 2

Set the argument in $\log ({(u)}^{6})$ less than or equal to $0$ to find where the expression is undefined.

${(u)}^{6} \leq 0$

Step 3

Solve for $u$ .

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Step 3.1

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

$\sqrt[6]{u^{6}} \leq \sqrt[6]{0}$

Step 3.2

Simplify the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Pull terms out from under the radical.

$| u | \leq \sqrt[6]{0}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\sqrt[6]{0}$ .

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Step 3.2.2.1.1

Rewrite $0$ as $0^{6}$ .

$| u | \leq \sqrt[6]{0^{6}}$

Step 3.2.2.1.2

Pull terms out from under the radical.

$| u | \leq | 0 |$

Step 3.2.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$| u | \leq 0$

Step 3.3

Write $| u | \leq 0$ as a piecewise.

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Step 3.3.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$u \geq 0$

Step 3.3.2

In the piece where $u$ is non-negative, remove the absolute value.

$u \leq 0$

Step 3.3.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$u < 0$

Step 3.3.4

In the piece where $u$ is negative, remove the absolute value and multiply by $- 1$ .

$- u \leq 0$

Step 3.3.5

Write as a piecewise.

${\begin{cases} u \leq 0 & u \geq 0 \\ - u \leq 0 & u < 0 \end{cases}$

Step 3.4

Find the intersection of $u \leq 0$ and $u \geq 0$ .

$u = 0$

Step 3.5

Solve $- u \leq 0$ when $u < 0$ .

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Step 3.5.1

Divide each term in $- u \leq 0$ by $- 1$ and simplify.

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Step 3.5.1.1

Divide each term in $- u \leq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- u}{- 1} \geq \frac{0}{- 1}$

Step 3.5.1.2

Simplify the left side.

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Step 3.5.1.2.1

Dividing two negative values results in a positive value.

$\frac{u}{1} \geq \frac{0}{- 1}$

Step 3.5.1.2.2

Divide $u$ by $1$ .

$u \geq \frac{0}{- 1}$

Step 3.5.1.3

Simplify the right side.

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Step 3.5.1.3.1

Divide $0$ by $- 1$ .

$u \geq 0$

Step 3.5.2

Find the intersection of $u \geq 0$ and $u < 0$ .

No solution

Step 3.6

Find the union of the solutions.

$u = 0$

Step 4

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 $0$ .

$u \leq 0$

$(- \infty, 0]$

Step 5