Solve for x natural log of x<=9

$\ln (x) \leq 9$

Step 1

Convert the inequality to an equality.

$\ln (x) = 9$

Step 2

Solve the equation.

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

$e^{\ln (x)} = e^{9}$

Rewrite $\ln (x) = 9$ 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 $b^{y} = x$ .

$e^{9} = x$

Rewrite the equation as $x = e^{9}$ .

$x = e^{9}$

Step 3

Find the domain of $\ln (x) - 9$ .

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

$x > 0$

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

$(0, \infty)$

Step 4

Use each root to create test intervals.

$x < 0$

$0 < x < e^{9}$

$x > e^{9}$

Step 5

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

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

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

$x = - 2$

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

$\ln (- 2) \leq 9$

Determine if the inequality is true.

The equation cannot be solved because it is undefined.

$Undefined$

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

False

Test a value on the interval $0 < x < e^{9}$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < e^{9}$ and see if this value makes the original inequality true.

$x = 2$

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

$\ln (2) \leq 9$

The left side $0.69314718$ is less than the right side $9$ , which means that the given statement is always true.

True

Test a value on the interval $x > e^{9}$ to see if it makes the inequality true.

Choose a value on the interval $x > e^{9}$ and see if this value makes the original inequality true.

$x = 8106$

Replace $x$ with $8106$ in the original inequality.

$\ln (8106) \leq 9$

The left side $9.0003598$ is greater than the right side $9$ , which means that the given statement is false.

False

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

$x < 0$ False

$0 < x < e^{9}$ True

$x > e^{9}$ False

$x < 0$ False

$0 < x < e^{9}$ True

$x > e^{9}$ False

Step 6

The solution consists of all of the true intervals.

$0 < x \leq e^{9}$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$0 < x \leq e^{9}$

Interval Notation:

$(0, e^{9}]$

Step 8

Solve for x natural log of x<=9

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