Solve the Inequality for x log base 4 of x<3

$\log_{4} (x) < 3$

Convert the inequality to an equality.

$\log_{4} (x) = 3$

Solve the equation.

Rewrite $\log_{4} (x) = 3$ 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 $b^{y} = x$ .

$4^{3} = x$

Solve for $x$ .

Rewrite the equation as $x = 4^{3}$ .

$x = 4^{3}$

Raise $4$ to the power of $3$ .

$x = 64$

Find the domain of $\log_{4} (x) - 3$ .

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

$x > 0$

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

$(0, \infty)$

Use each root to create test intervals.

$x < 0$

$0 < x < 64$

$x > 64$

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.

$\log_{4} (- 2) < 3$

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 $0 < x < 64$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < 64$ and see if this value makes the original inequality true.

$x = 2$

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

$\log_{4} (2) < 3$

The left side $0.5$ is less than the right side $3$ , which means that the given statement is always true.

True

Test a value on the interval $x > 64$ to see if it makes the inequality true.

Choose a value on the interval $x > 64$ and see if this value makes the original inequality true.

$x = 66$

Replace $x$ with $66$ in the original inequality.

$\log_{4} (66) < 3$

The left side $3.02219705$ is not less than the right side $3$ , which means that the given statement is false.

False

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

$x < 0$ False

$0 < x < 64$ True

$x > 64$ False

$x < 0$ False

$0 < x < 64$ True

$x > 64$ False

The solution consists of all of the true intervals.

$0 < x < 64$

The result can be shown in multiple forms.

Inequality Form:

$0 < x < 64$

Interval Notation:

$(0, 64)$

Solve the Inequality for x log base 4 of x<3

Related Questions

Support LCMGCF.com by disabling your adblocker.