Algebra Examples

$\log_{256} (x) < \frac{1}{4}$

Step 1

Convert the inequality to an equality.

$\log_{256} (x) = \frac{1}{4}$

Step 2

Solve the equation.

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Step 2.1

Rewrite $\log_{256} (x) = \frac{1}{4}$ 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 $b^{y} = x$ .

$256^{\frac{1}{4}} = x$

Step 2.2

Solve for $x$ .

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Step 2.2.1

Rewrite the equation as $x = 256^{\frac{1}{4}}$ .

$x = 256^{\frac{1}{4}}$

Step 2.2.2

Simplify $256^{\frac{1}{4}}$ .

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Step 2.2.2.1

Simplify the expression.

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Step 2.2.2.1.1

Rewrite $256$ as $4^{4}$ .

$x = {(4^{4})}^{\frac{1}{4}}$

Step 2.2.2.1.2

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

$x = 4^{4 (\frac{1}{4})}$

Step 2.2.2.2

Cancel the common factor of $4$ .

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Step 2.2.2.2.1

Cancel the common factor.

$x = 4^{4 (\frac{1}{4})}$

Step 2.2.2.2.2

Rewrite the expression.

$x = 4^{1}$

Step 2.2.2.3

Evaluate the exponent.

$x = 4$

Step 3

Find the domain of $\log_{256} (x) - \frac{1}{4}$ .

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Step 3.1

Set the argument in $\log_{256} (x)$ greater than $0$ to find where the expression is defined.

$x > 0$

Step 3.2

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

$(0, \infty)$

Step 4

Use each root to create test intervals.

$x < 0$

$0 < x < 4$

$x > 4$

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

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Step 5.1.1

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

$x = - 2$

Step 5.1.2

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

$\log_{256} (- 2) < \frac{1}{4}$

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 $0 < x < 4$ to see if it makes the inequality true.

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Step 5.2.1

Choose a value on the interval $0 < x < 4$ and see if this value makes the original inequality true.

$x = 2$

Step 5.2.2

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

$\log_{256} (2) < \frac{1}{4}$

Step 5.2.3

The left side $0.125$ is less than the right side $0.25$ , which means that the given statement is always true.

True

Step 5.3

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

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Step 5.3.1

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

$x = 6$

Step 5.3.2

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

$\log_{256} (6) < \frac{1}{4}$

Step 5.3.3

The left side $0.32312031$ is not less than the right side $0.25$ , which means that the given statement is false.

False

Step 5.4

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

$x < 0$ False

$0 < x < 4$ True

$x > 4$ False

$x < 0$ False

$0 < x < 4$ True

$x > 4$ False

Step 6

The solution consists of all of the true intervals.

$0 < x < 4$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$0 < x < 4$

Interval Notation:

$(0, 4)$

Step 8