Precalculus Examples

$\log_{3} (\frac{m^{2} π}{\sqrt[5]{3 π - 1}})$

Step 1

Rewrite $\log_{3} (\frac{m^{2} π}{\sqrt[5]{3 π - 1}})$ as $\log_{3} (m^{2} π) - \log_{3} (\sqrt[5]{3 π - 1})$ .

$\log_{3} (m^{2} π) - \log_{3} (\sqrt[5]{3 π - 1})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{3 π - 1}$ as ${(3 π - 1)}^{\frac{1}{5}}$ .

$\log_{3} (m^{2} π) - \log_{3} ({(3 π - 1)}^{\frac{1}{5}})$

Step 3

Expand $\log_{3} ({(3 π - 1)}^{\frac{1}{5}})$ by moving $\frac{1}{5}$ outside the logarithm.

$\log_{3} (m^{2} π) - (\frac{1}{5} \log_{3} (3 π - 1))$

Step 4

Combine $\frac{1}{5}$ and $\log_{3} (3 π - 1)$ .

$\log_{3} (m^{2} π) - \frac{\log_{3} (3 π - 1)}{5}$

Step 5

To write $\log_{3} (m^{2} π)$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\log_{3} (m^{2} π) \cdot \frac{5}{5} - \frac{\log_{3} (3 π - 1)}{5}$

Step 6

Combine $\log_{3} (m^{2} π)$ and $\frac{5}{5}$ .

$\frac{\log_{3} (m^{2} π) \cdot 5}{5} - \frac{\log_{3} (3 π - 1)}{5}$

Step 7

Combine the numerators over the common denominator.

$\frac{\log_{3} (m^{2} π) \cdot 5 - \log_{3} (3 π - 1)}{5}$

Step 8

Rewrite $\log_{3} (m^{2} π)$ as $\log_{3} (m^{2}) + \log_{3} (π)$ .

$\frac{(\log_{3} (m^{2}) + \log_{3} (π)) \cdot 5 - \log_{3} (3 π - 1)}{5}$

Step 9

Expand $\log_{3} (m^{2})$ by moving $2$ outside the logarithm.

$\frac{(2 \log_{3} (m) + \log_{3} (π)) \cdot 5 - \log_{3} (3 π - 1)}{5}$

Step 10

Simplify the numerator.

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Step 10.1

Apply the distributive property.

$\frac{2 \log_{3} (m) \cdot 5 + \log_{3} (π) \cdot 5 - \log_{3} (3 π - 1)}{5}$

Step 10.2

Multiply $5$ by $2$ .

$\frac{10 \log_{3} (m) + \log_{3} (π) \cdot 5 - \log_{3} (3 π - 1)}{5}$

Step 10.3

Move $5$ to the left of $\log_{3} (π)$ .

$\frac{10 \log_{3} (m) + 5 \log_{3} (π) - \log_{3} (3 π - 1)}{5}$