Trigonometry Examples

$- 1 \cdot \log_{5} (x - 4) > 0$

Step 1

Convert the inequality to an equality.

$- 1 \cdot \log_{5} (x - 4) = 0$

Step 2

Solve the equation.

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Step 2.1

Divide each term in $- 1 \cdot \log_{5} (x - 4) = 0$ by $- 1$ and simplify.

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Step 2.1.1

Divide each term in $- 1 \cdot \log_{5} (x - 4) = 0$ by $- 1$ .

$\frac{- 1 \cdot \log_{5} (x - 4)}{- 1} = \frac{0}{- 1}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Dividing two negative values results in a positive value.

$\frac{\log_{5} (x - 4)}{1} = \frac{0}{- 1}$

Step 2.1.2.2

Divide $\log_{5} (x - 4)$ by $1$ .

$\log_{5} (x - 4) = \frac{0}{- 1}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $0$ by $- 1$ .

$\log_{5} (x - 4) = 0$

Step 2.2

Rewrite $\log_{5} (x - 4) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$5^{0} = x - 4$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Rewrite the equation as $x - 4 = 5^{0}$ .

$x - 4 = 5^{0}$

Step 2.3.2

Anything raised to $0$ is $1$ .

$x - 4 = 1$

Step 2.3.3

Move all terms not containing $x$ to the right side of the equation.

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Step 2.3.3.1

Add $4$ to both sides of the equation.

$x = 1 + 4$

Step 2.3.3.2

Add $1$ and $4$ .

$x = 5$

Step 3

Find the domain of $- 1 \cdot \log_{5} (x - 4)$ .

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Step 3.1

Set the argument in $\log_{5} (x - 4)$ greater than $0$ to find where the expression is defined.

$x - 4 > 0$

Step 3.2

Add $4$ to both sides of the inequality.

$x > 4$

Step 3.3

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

$(4, \infty)$

Step 4

Use each root to create test intervals.

$x < 4$

$4 < x < 5$

$x > 5$

Step 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 5.1

Test a value on the interval $x < 4$ to see if it makes the inequality true.

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Step 5.1.1

Choose a value on the interval $x < 4$ and see if this value makes the original inequality true.

$x = 0$

Step 5.1.2

Replace $x$ with $0$ in the original inequality.

$- 1 \cdot \log_{5} ((0) - 4) > 0$

Step 5.1.3

Determine if the inequality is true.

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Step 5.1.3.1

The equation cannot be solved because it is undefined.

$Undefined$

Step 5.1.3.2

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

False

Step 5.2

Test a value on the interval $4 < x < 5$ to see if it makes the inequality true.

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Step 5.2.1

Choose a value on the interval $4 < x < 5$ and see if this value makes the original inequality true.

$x = 4.5$

Step 5.2.2

Replace $x$ with $4.5$ in the original inequality.

$- 1 \cdot \log_{5} ((4.5) - 4) > 0$

Step 5.2.3

The left side $0.43067655$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 5.3

Test a value on the interval $x > 5$ to see if it makes the inequality true.

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Step 5.3.1

Choose a value on the interval $x > 5$ and see if this value makes the original inequality true.

$x = 8$

Step 5.3.2

Replace $x$ with $8$ in the original inequality.

$- 1 \cdot \log_{5} ((8) - 4) > 0$

Step 5.3.3

The left side $- 0.86135311$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 5.4

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

$x < 4$ False

$4 < x < 5$ True

$x > 5$ False

$x < 4$ False

$4 < x < 5$ True

$x > 5$ False

Step 6

The solution consists of all of the true intervals.

$4 < x < 5$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$4 < x < 5$

Interval Notation:

$(4, 5)$

Step 8