Calculus Examples

$\ln (x - 2) < 0$

Step 1

Convert the inequality to an equality.

$\ln (x - 2) = 0$

Step 2

Solve the equation.

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Step 2.1

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (x - 2)} = e^{0}$

Step 2.2

Rewrite $\ln (x - 2) = 0$ 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 $b^{y} = x$ .

$e^{0} = x - 2$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Rewrite the equation as $x - 2 = e^{0}$ .

$x - 2 = e^{0}$

Step 2.3.2

Anything raised to $0$ is $1$ .

$x - 2 = 1$

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 $2$ to both sides of the equation.

$x = 1 + 2$

Step 2.3.3.2

Add $1$ and $2$ .

$x = 3$

Step 3

Find the domain of $\ln (x - 2)$ .

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Step 3.1

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

$x - 2 > 0$

Step 3.2

Add $2$ to both sides of the inequality.

$x > 2$

Step 3.3

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

$(2, \infty)$

Step 4

Use each root to create test intervals.

$x < 2$

$2 < x < 3$

$x > 3$

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 $x < 2$ to see if it makes the inequality true.

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Step 5.1.1

Choose a value on the interval $x < 2$ and see if this value makes the original inequality true.

$x = 0$

Step 5.1.2

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

$\ln ((0) - 2) < 0$

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 $2 < x < 3$ to see if it makes the inequality true.

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Step 5.2.1

Choose a value on the interval $2 < x < 3$ and see if this value makes the original inequality true.

$x = 2.5$

Step 5.2.2

Replace $x$ with $2.5$ in the original inequality.

$\ln ((2.5) - 2) < 0$

Step 5.2.3

The left side $- 0.69314718$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 5.3

Test a value on the interval $x > 3$ to see if it makes the inequality true.

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Step 5.3.1

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

Step 5.3.2

Replace $x$ with $6$ in the original inequality.

$\ln ((6) - 2) < 0$

Step 5.3.3

The left side $1.38629436$ is not less 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.

$x < 2$ False

$2 < x < 3$ True

$x > 3$ False

$x < 2$ False

$2 < x < 3$ True

$x > 3$ False

Step 6

The solution consists of all of the true intervals.

$2 < x < 3$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$2 < x < 3$

Interval Notation:

$(2, 3)$

Step 8