Precalculus Examples

$4^{(\frac{\log_{4} (10000^{1})}{\log_{4} (\frac{4}{5})}) - 2}$

Step 1

Evaluate the exponent.

$4^{\frac{\log_{4} (10000)}{\log_{4} (\frac{4}{5})} - 2}$

Step 2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{\log_{4} (\frac{4}{5})}{\log_{4} (\frac{4}{5})}$ .

$4^{\frac{\log_{4} (10000)}{\log_{4} (\frac{4}{5})} - 2 \cdot \frac{\log_{4} (\frac{4}{5})}{\log_{4} (\frac{4}{5})}}$

Step 3

Combine fractions.

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Step 3.1

Combine $- 2$ and $\frac{\log_{4} (\frac{4}{5})}{\log_{4} (\frac{4}{5})}$ .

$4^{\frac{\log_{4} (10000)}{\log_{4} (\frac{4}{5})} + \frac{- 2 \log_{4} (\frac{4}{5})}{\log_{4} (\frac{4}{5})}}$

Step 3.2

Combine the numerators over the common denominator.

$4^{\frac{\log_{4} (10000) - 2 \log_{4} (\frac{4}{5})}{\log_{4} (\frac{4}{5})}}$

Step 4

Simplify the numerator.

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Step 4.1

Simplify $- 2 \log_{4} (\frac{4}{5})$ by moving $2$ inside the logarithm.

$4^{\frac{\log_{4} (10000) - \log_{4} ({(\frac{4}{5})}^{2})}{\log_{4} (\frac{4}{5})}}$

Step 4.2

Apply the product rule to $\frac{4}{5}$ .

$4^{\frac{\log_{4} (10000) - \log_{4} (\frac{4^{2}}{5^{2}})}{\log_{4} (\frac{4}{5})}}$

Step 4.3

Raise $4$ to the power of $2$ .

$4^{\frac{\log_{4} (10000) - \log_{4} (\frac{16}{5^{2}})}{\log_{4} (\frac{4}{5})}}$

Step 4.4

Raise $5$ to the power of $2$ .

$4^{\frac{\log_{4} (10000) - \log_{4} (\frac{16}{25})}{\log_{4} (\frac{4}{5})}}$

Step 4.5

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$4^{\frac{\log_{4} (\frac{10000}{\frac{16}{25}})}{\log_{4} (\frac{4}{5})}}$

Step 4.6

Multiply the numerator by the reciprocal of the denominator.

$4^{\frac{\log_{4} (10000 (\frac{25}{16}))}{\log_{4} (\frac{4}{5})}}$

Step 4.7

Cancel the common factor of $16$ .

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Step 4.7.1

Factor $16$ out of $10000$ .

$4^{\frac{\log_{4} (16 (625) \frac{25}{16})}{\log_{4} (\frac{4}{5})}}$

Step 4.7.2

Cancel the common factor.

$4^{\frac{\log_{4} (16 \cdot 625 \frac{25}{16})}{\log_{4} (\frac{4}{5})}}$

Step 4.7.3

Rewrite the expression.

$4^{\frac{\log_{4} (625 \cdot 25)}{\log_{4} (\frac{4}{5})}}$

Step 4.8

Multiply $625$ by $25$ .

$4^{\frac{\log_{4} (15625)}{\log_{4} (\frac{4}{5})}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$4^{\frac{\log_{4} (15625)}{\log_{4} (\frac{4}{5})}}$

Decimal Form:

$8.82287774 \cdot 10^{- 27}$