Solve for x log base 125 of x<1/3

$\log_{125} (x) < \frac{1}{3}$

Convert the inequality to an equality.

$\log_{125} (x) = \frac{1}{3}$

Solve the equation.

Rewrite $\log_{125} (x) = \frac{1}{3}$ 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 $b^{y} = x$ .

$125^{\frac{1}{3}} = x$

Solve for $x$ .

Rewrite the equation as $x = 125^{\frac{1}{3}}$ .

$x = 125^{\frac{1}{3}}$

Simplify $125^{\frac{1}{3}}$ .

Simplify the expression.

Rewrite $125$ as $5^{3}$ .

$x = {(5^{3})}^{\frac{1}{3}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = 5^{3 (\frac{1}{3})}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x = 5^{3 (\frac{1}{3})}$

Rewrite the expression.

$x = 5^{1}$

Evaluate the exponent.

$x = 5$

Find the domain of $\log_{125} (x) - \frac{1}{3}$ .

Set the argument in $\log_{125} (x)$ 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.

$x < 0$

$0 < x < 5$

$x > 5$

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.

$\log_{125} (- 2) < \frac{1}{3}$

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 $0 < x < 5$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < 5$ and see if this value makes the original inequality true.

$x = 2$

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

$\log_{125} (2) < \frac{1}{3}$

The left side $0.14355885$ is less than the right side $0.\overline{3}$ , which means that the given statement is always true.

True

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

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

$x = 8$

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

$\log_{125} (8) < \frac{1}{3}$

The left side $0.43067655$ is not less than the right side $0.\overline{3}$ , which means that the given statement is false.

False

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

$x < 0$ False

$0 < x < 5$ True

$x > 5$ False

$x < 0$ False

$0 < x < 5$ True

$x > 5$ False

The solution consists of all of the true intervals.

$0 < x < 5$

The result can be shown in multiple forms.

Inequality Form:

$0 < x < 5$

Interval Notation:

$(0, 5)$

Solve for x log base 125 of x<1/3

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