Algebra Examples

$\log_{32} (x) > \frac{1}{5}$

Step 1

Convert the inequality to an equality.

$\log_{32} (x) = \frac{1}{5}$

Step 2

Solve the equation.

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Step 2.1

Rewrite $\log_{32} (x) = \frac{1}{5}$ 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 $b^{y} = x$ .

$32^{\frac{1}{5}} = x$

Step 2.2

Solve for $x$ .

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Step 2.2.1

Rewrite the equation as $x = 32^{\frac{1}{5}}$ .

$x = 32^{\frac{1}{5}}$

Step 2.2.2

Simplify $32^{\frac{1}{5}}$ .

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Step 2.2.2.1

Simplify the expression.

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Step 2.2.2.1.1

Rewrite $32$ as $2^{5}$ .

$x = {(2^{5})}^{\frac{1}{5}}$

Step 2.2.2.1.2

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

$x = 2^{5 (\frac{1}{5})}$

Step 2.2.2.2

Cancel the common factor of $5$ .

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Step 2.2.2.2.1

Cancel the common factor.

$x = 2^{5 (\frac{1}{5})}$

Step 2.2.2.2.2

Rewrite the expression.

$x = 2^{1}$

Step 2.2.2.3

Evaluate the exponent.

$x = 2$

Step 3

Find the domain of $\log_{32} (x) - \frac{1}{5}$ .

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Step 3.1

Set the argument in $\log_{32} (x)$ 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

The solution consists of all of the true intervals.

$x > 2$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x > 2$

Interval Notation:

$(2, \infty)$

Step 6