Convert the inequality to an equality.

Rewrite <math><mstyle displaystyle="true"><msub><mi>log</mi><mrow><mn>256</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> in exponential form using the definition of a logarithm. If <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> and <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> are positive real numbers and <math><mstyle displaystyle="true"><mi>b</mi><mo>≠</mo><mn>1</mn></mstyle></math> , then <math><mstyle displaystyle="true"><msub><mi>log</mi><mrow><mi>b</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>y</mi></mstyle></math> is equivalent to <math><mstyle displaystyle="true"><msup><mrow><mi>b</mi></mrow><mrow><mi>y</mi></mrow></msup><mo>=</mo><mi>x</mi></mstyle></math> .

Solve for <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Rewrite the equation as <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><msup><mrow><mn>256</mn></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></mstyle></math> .

Simplify <math><mstyle displaystyle="true"><msup><mrow><mn>256</mn></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></mstyle></math> .

Simplify the expression.

Rewrite <math><mstyle displaystyle="true"><mn>256</mn></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>4</mn></mrow><mrow><mn>4</mn></mrow></msup></mstyle></math> .

Apply the power rule and multiply exponents, <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msup></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Evaluate the exponent.

Set the argument in <math><mstyle displaystyle="true"><msub><mi>log</mi><mrow><mn>256</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> greater than <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> to find where the expression is defined.

The domain is all values of <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> that make the expression defined.

Use each root to create test intervals.

Test a value on the interval <math><mstyle displaystyle="true"><mi>x</mi><mo><</mo><mn>0</mn></mstyle></math> to see if it makes the inequality true.

Choose a value on the interval <math><mstyle displaystyle="true"><mi>x</mi><mo><</mo><mn>0</mn></mstyle></math> and see if this value makes the original inequality true.

Replace <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> in the original inequality.

Determine if the inequality is true.

The equation cannot be solved because it is undefined.

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

False

False

False

Test a value on the interval <math><mstyle displaystyle="true"><mn>0</mn><mo><</mo><mi>x</mi><mo><</mo><mn>4</mn></mstyle></math> to see if it makes the inequality true.

Choose a value on the interval <math><mstyle displaystyle="true"><mn>0</mn><mo><</mo><mi>x</mi><mo><</mo><mn>4</mn></mstyle></math> and see if this value makes the original inequality true.

Replace <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> in the original inequality.

The left side <math><mstyle displaystyle="true"><mn>0.125</mn></mstyle></math> is less than the right side <math><mstyle displaystyle="true"><mn>0.25</mn></mstyle></math> , which means that the given statement is always true.

True

True

Test a value on the interval <math><mstyle displaystyle="true"><mi>x</mi><mo>></mo><mn>4</mn></mstyle></math> to see if it makes the inequality true.

Choose a value on the interval <math><mstyle displaystyle="true"><mi>x</mi><mo>></mo><mn>4</mn></mstyle></math> and see if this value makes the original inequality true.

Replace <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> in the original inequality.

The left side <math><mstyle displaystyle="true"><mn>0.32312031</mn></mstyle></math> is not less than the right side <math><mstyle displaystyle="true"><mn>0.25</mn></mstyle></math> , which means that the given statement is false.

False

False

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

The solution consists of all of the true intervals.

The result can be shown in multiple forms.

Inequality Form:

Interval Notation:

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