Algebra Examples

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

Step 1

Solve $0 < x < 256$ .

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Step 1.1

Convert the inequality to an equality.

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

$4^{4} = x$

Step 1.2.2

Solve for $x$ .

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Step 1.2.2.1

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

$x = 4^{4}$

Step 1.2.2.2

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

$x = 256$

Step 1.3

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

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Step 1.3.1

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

$x > 0$

Step 1.3.2

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

$(0, \infty)$

Step 1.4

Use each root to create test intervals.

$x < 0$

$0 < x < 256$

$x > 256$

Step 1.5

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 1.5.1

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

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Step 1.5.1.1

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

$x = - 2$

Step 1.5.1.2

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

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

Step 1.5.1.3

Determine if the inequality is true.

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Step 1.5.1.3.1

The equation cannot be solved because it is undefined.

$Undefined$

Step 1.5.1.3.2

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

False

Step 1.5.2

Test a value on the interval $0 < x < 256$ to see if it makes the inequality true.

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Step 1.5.2.1

Choose a value on the interval $0 < x < 256$ and see if this value makes the original inequality true.

$x = 2$

Step 1.5.2.2

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

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

Step 1.5.2.3

The left side $0.5$ is less than the right side $4$ , which means that the given statement is always true.

True

Step 1.5.3

Test a value on the interval $x > 256$ to see if it makes the inequality true.

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Step 1.5.3.1

Choose a value on the interval $x > 256$ and see if this value makes the original inequality true.

$x = 258$

Step 1.5.3.2

Replace $x$ with $258$ in the original inequality.

$\log_{4} (258) < 4$

Step 1.5.3.3

The left side $4.00561362$ is not less than the right side $4$ , which means that the given statement is false.

False

Step 1.5.4

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

$x < 0$ False

$0 < x < 256$ True

$x > 256$ False

$x < 0$ False

$0 < x < 256$ True

$x > 256$ False

Step 1.6

The solution consists of all of the true intervals.

$0 < x < 256$

Step 2

Use the inequality $0 < x < 256$ to build the set notation.

${x | 0 < x < 256}$

Step 3