Solve for b log base 256 of b>1/4

$\log_{256} (b) > \frac{1}{4}$

Step 1

Convert the inequality to an equality.

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

Step 2

Solve the equation.

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

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

Solve for $b$ .

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

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

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

Simplify the expression.

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

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

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

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

$b = 4^{1}$

Evaluate the exponent.

$b = 4$

Step 3

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

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

$b > 0$

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

$(0, \infty)$

Step 4

The solution consists of all of the true intervals.

$b > 4$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$b > 4$

Interval Notation:

$(4, \infty)$

Step 6

Solve for b log base 256 of b>1/4

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