Precalculus Examples

$\log_{3} (\frac{y^{3}}{100 \sqrt[3]{x}})$

Step 1

Rewrite $\log_{3} (\frac{y^{3}}{100 \sqrt[3]{x}})$ as $\log_{3} (y^{3}) - \log_{3} (100 \sqrt[3]{x})$ .

$\log_{3} (y^{3}) - \log_{3} (100 \sqrt[3]{x})$

Step 2

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

$\log_{3} (y^{3}) - \log_{3} (100 x^{\frac{1}{3}})$

Step 3

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

$3 \log_{3} (y) - \log_{3} (100 x^{\frac{1}{3}})$

Step 4

Rewrite $\log_{3} (100 x^{\frac{1}{3}})$ as $\log_{3} (100) + \log_{3} (x^{\frac{1}{3}})$ .

$3 \log_{3} (y) - (\log_{3} (100) + \log_{3} (x^{\frac{1}{3}}))$

Step 5

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

$3 \log_{3} (y) - (\log_{3} (100) + \frac{1}{3} \log_{3} (x))$

Step 6

Simplify each term.

Tap for more steps...

Step 6.1

Simplify each term.

Tap for more steps...

Step 6.1.1

Rewrite $\log_{3} (100)$ as $\log_{3} (2^{2} \cdot 5^{2})$ .

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

Step 6.1.2

Rewrite $\log_{3} (2^{2} \cdot 5^{2})$ as $\log_{3} (2^{2}) + \log_{3} (5^{2})$ .

$3 \log_{3} (y) - (\log_{3} (2^{2}) + \log_{3} (5^{2}) + \frac{1}{3} \log_{3} (x))$

Step 6.1.3

Simplify each term.

Tap for more steps...

Step 6.1.3.1

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

$3 \log_{3} (y) - (2 \log_{3} (2) + \log_{3} (5^{2}) + \frac{1}{3} \log_{3} (x))$

Step 6.1.3.2

Expand $\log_{3} (5^{2})$ by moving $2$ outside the logarithm.

$3 \log_{3} (y) - (2 \log_{3} (2) + 2 \log_{3} (5) + \frac{1}{3} \log_{3} (x))$

Step 6.1.4

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

$3 \log_{3} (y) - (2 \log_{3} (2) + 2 \log_{3} (5) + \frac{\log_{3} (x)}{3})$

Step 6.2

Apply the distributive property.

$3 \log_{3} (y) - (2 \log_{3} (2)) - (2 \log_{3} (5)) - \frac{\log_{3} (x)}{3}$

Step 6.3

Simplify.

Tap for more steps...

Step 6.3.1

Multiply $2$ by $- 1$ .

$3 \log_{3} (y) - 2 \log_{3} (2) - (2 \log_{3} (5)) - \frac{\log_{3} (x)}{3}$

Step 6.3.2

Multiply $2$ by $- 1$ .

$3 \log_{3} (y) - 2 \log_{3} (2) - 2 \log_{3} (5) - \frac{\log_{3} (x)}{3}$