Calculus Examples

$\log_{3} (x^{3} y^{2} \sqrt{z})$

Step 1

Rewrite $\log_{3} (x^{3} y^{2} \sqrt{z})$ as $\log_{3} (x^{3} y^{2}) + \log_{3} (\sqrt{z})$ .

$\log_{3} (x^{3} y^{2}) + \log_{3} (\sqrt{z})$

Step 2

Rewrite $\log_{3} (x^{3} y^{2})$ as $\log_{3} (x^{3}) + \log_{3} (y^{2})$ .

$\log_{3} (x^{3}) + \log_{3} (y^{2}) + \log_{3} (\sqrt{z})$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z}$ as $z^{\frac{1}{2}}$ .

$\log_{3} (x^{3}) + \log_{3} (y^{2}) + \log_{3} (z^{\frac{1}{2}})$

Step 4

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

$3 \log_{3} (x) + \log_{3} (y^{2}) + \log_{3} (z^{\frac{1}{2}})$

Step 5

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

$3 \log_{3} (x) + 2 \log_{3} (y) + \log_{3} (z^{\frac{1}{2}})$

Step 6

Expand $\log_{3} (z^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$3 \log_{3} (x) + 2 \log_{3} (y) + \frac{1}{2} \log_{3} (z)$

Step 7

Combine $\frac{1}{2}$ and $\log_{3} (z)$ .

$3 \log_{3} (x) + 2 \log_{3} (y) + \frac{\log_{3} (z)}{2}$