Precalculus Examples

$\log_{3} (7 x^{3} \sqrt{y})$

Step 1

Rewrite $\log_{3} (7 x^{3} \sqrt{y})$ as $\log_{3} (7 x^{3}) + \log_{3} (\sqrt{y})$ .

$\log_{3} (7 x^{3}) + \log_{3} (\sqrt{y})$

Step 2

Rewrite $\log_{3} (7 x^{3})$ as $\log_{3} (7) + \log_{3} (x^{3})$ .

$\log_{3} (7) + \log_{3} (x^{3}) + \log_{3} (\sqrt{y})$

Step 3

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

$\log_{3} (7) + \log_{3} (x^{3}) + \log_{3} (y^{\frac{1}{2}})$

Step 4

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

$\log_{3} (7) + 3 \log_{3} (x) + \log_{3} (y^{\frac{1}{2}})$

Step 5

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

$\log_{3} (7) + 3 \log_{3} (x) + \frac{1}{2} \log_{3} (y)$

Step 6

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

$\log_{3} (7) + 3 \log_{3} (x) + \frac{\log_{3} (y)}{2}$