Precalculus Examples

$\log_{4} (\frac{1}{\sqrt[5]{4}})$

Step 1

Multiply $\frac{1}{\sqrt[5]{4}}$ by $\frac{{\sqrt[5]{4}}^{4}}{{\sqrt[5]{4}}^{4}}$ .

$\log_{4} (\frac{1}{\sqrt[5]{4}} \cdot \frac{{\sqrt[5]{4}}^{4}}{{\sqrt[5]{4}}^{4}})$

Step 2

Combine and simplify the denominator.

Tap for more steps...

Step 2.1

Multiply $\frac{1}{\sqrt[5]{4}}$ by $\frac{{\sqrt[5]{4}}^{4}}{{\sqrt[5]{4}}^{4}}$ .

$\log_{4} (\frac{{\sqrt[5]{4}}^{4}}{\sqrt[5]{4} {\sqrt[5]{4}}^{4}})$

Step 2.2

Raise $\sqrt[5]{4}$ to the power of $1$ .

$\log_{4} (\frac{{\sqrt[5]{4}}^{4}}{{\sqrt[5]{4}}^{1} {\sqrt[5]{4}}^{4}})$

Step 2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{4} (\frac{{\sqrt[5]{4}}^{4}}{{\sqrt[5]{4}}^{1 + 4}})$

Step 2.4

Add $1$ and $4$ .

$\log_{4} (\frac{{\sqrt[5]{4}}^{4}}{{\sqrt[5]{4}}^{5}})$

Step 2.5

Rewrite ${\sqrt[5]{4}}^{5}$ as $4$ .

Tap for more steps...

Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{4}$ as $4^{\frac{1}{5}}$ .

$\log_{4} (\frac{{\sqrt[5]{4}}^{4}}{{(4^{\frac{1}{5}})}^{5}})$

Step 2.5.2

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

$\log_{4} (\frac{{\sqrt[5]{4}}^{4}}{4^{\frac{1}{5} \cdot 5}})$

Step 2.5.3

Combine $\frac{1}{5}$ and $5$ .

$\log_{4} (\frac{{\sqrt[5]{4}}^{4}}{4^{\frac{5}{5}}})$

Step 2.5.4

Cancel the common factor of $5$ .

Tap for more steps...

Step 2.5.4.1

Cancel the common factor.

$\log_{4} (\frac{{\sqrt[5]{4}}^{4}}{4^{\frac{5}{5}}})$

Step 2.5.4.2

Rewrite the expression.

$\log_{4} (\frac{{\sqrt[5]{4}}^{4}}{4^{1}})$

Step 2.5.5

Evaluate the exponent.

$\log_{4} (\frac{{\sqrt[5]{4}}^{4}}{4})$

Step 3

Rewrite $\frac{{\sqrt[5]{4}}^{4}}{4}$ as ${\sqrt[5]{4}}^{4} \cdot 4^{- 1}$ .

$\log_{4} ({\sqrt[5]{4}}^{4} \cdot 4^{- 1})$

Step 4

Rewrite $\log_{4} ({\sqrt[5]{4}}^{4} \cdot 4^{- 1})$ as $\log_{4} ({\sqrt[5]{4}}^{4}) + \log_{4} (4^{- 1})$ .

$\log_{4} ({\sqrt[5]{4}}^{4}) + \log_{4} (4^{- 1})$

Step 5

Use logarithm rules to move $- 1$ out of the exponent.

$\log_{4} ({\sqrt[5]{4}}^{4}) - \log_{4} (4)$

Step 6

Logarithm base $4$ of $4$ is $1$ .

$\log_{4} ({\sqrt[5]{4}}^{4}) - 1 \cdot 1$

Step 7

Multiply $- 1$ by $1$ .

$\log_{4} ({\sqrt[5]{4}}^{4}) - 1$

Step 8

Expand $\log_{4} ({\sqrt[5]{4}}^{4})$ by moving $4$ outside the logarithm.

$4 \log_{4} (\sqrt[5]{4}) - 1$

Step 9

Simplify each term.

Tap for more steps...

Step 9.1

Rewrite $\log_{4} (\sqrt[5]{4})$ using the change of base formula.

Tap for more steps...

Step 9.1.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$4 \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} - 1$

Step 9.1.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$4 \frac{\log (\sqrt[5]{4})}{\log (4)} - 1$

Step 9.2

Multiply $4 \frac{\log (\sqrt[5]{4})}{\log (4)}$ .

Tap for more steps...

Step 9.2.1

Combine $4$ and $\frac{\log (\sqrt[5]{4})}{\log (4)}$ .

$\frac{4 \log (\sqrt[5]{4})}{\log (4)} - 1$

Step 9.2.2

Simplify $4 \log (\sqrt[5]{4})$ by moving $4$ inside the logarithm.

$\frac{\log ({\sqrt[5]{4}}^{4})}{\log (4)} - 1$

Step 9.3

Simplify the numerator.

Tap for more steps...

Step 9.3.1

Rewrite ${\sqrt[5]{4}}^{4}$ as $\sqrt[5]{4^{4}}$ .

$\frac{\log (\sqrt[5]{4^{4}})}{\log (4)} - 1$

Step 9.3.2

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

$\frac{\log (\sqrt[5]{256})}{\log (4)} - 1$

Step 9.3.3

Rewrite $256$ as $2^{5} \cdot 8$ .

Tap for more steps...

Step 9.3.3.1

Factor $32$ out of $256$ .

$\frac{\log (\sqrt[5]{32 (8)})}{\log (4)} - 1$

Step 9.3.3.2

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

$\frac{\log (\sqrt[5]{2^{5} \cdot 8})}{\log (4)} - 1$

Step 9.3.4

Pull terms out from under the radical.

$\frac{\log (2 \sqrt[5]{8})}{\log (4)} - 1$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (2 \sqrt[5]{8})}{\log (4)} - 1$

Decimal Form:

$- 0.2$