Precalculus Examples

$\log (\frac{y^{3} ({(\sqrt[3]{z + 3})}^{5})}{x})$

Step 1

Rewrite $\log (\frac{y^{3} ({(\sqrt[3]{z + 3})}^{5})}{x})$ as $\log (y^{3} ({(\sqrt[3]{z + 3})}^{5})) - \log (x)$ .

$\log (y^{3} ({(\sqrt[3]{z + 3})}^{5})) - \log (x)$

Step 2

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

$\log (y^{3} {({(z + 3)}^{\frac{1}{3}})}^{5}) - \log (x)$

Step 3

Rewrite $\log (y^{3} {({(z + 3)}^{\frac{1}{3}})}^{5})$ as $\log (y^{3}) + \log ({({(z + 3)}^{\frac{1}{3}})}^{5})$ .

$\log (y^{3}) + \log ({({(z + 3)}^{\frac{1}{3}})}^{5}) - \log (x)$

Step 4

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

$3 \log (y) + \log ({({(z + 3)}^{\frac{1}{3}})}^{5}) - \log (x)$

Step 5

Expand $\log ({({(z + 3)}^{\frac{1}{3}})}^{5})$ by moving $5$ outside the logarithm.

$3 \log (y) + 5 \log ({(z + 3)}^{\frac{1}{3}}) - \log (x)$

Step 6

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

$3 \log (y) + 5 (\frac{1}{3} \log (z + 3)) - \log (x)$

Step 7

Combine $\frac{1}{3}$ and $5$ .

$3 \log (y) + \frac{5}{3} \log (z + 3) - \log (x)$

Step 8

Combine $\frac{5}{3}$ and $\log (z + 3)$ .

$3 \log (y) + \frac{5 \log (z + 3)}{3} - \log (x)$

Step 9

To write $3 \log (y)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 \log (y) \cdot \frac{3}{3} + \frac{5 \log (z + 3)}{3} - \log (x)$

Step 10

Combine $3 \log (y)$ and $\frac{3}{3}$ .

$\frac{3 \log (y) \cdot 3}{3} + \frac{5 \log (z + 3)}{3} - \log (x)$

Step 11

Combine the numerators over the common denominator.

$\frac{3 \log (y) \cdot 3 + 5 \log (z + 3)}{3} - \log (x)$

Step 12

Multiply $3$ by $3$ .

$\frac{9 \log (y) + 5 \log (z + 3)}{3} - \log (x)$