Precalculus Examples

$\log_{7} (49 x^{3} \sqrt[5]{y^{2}})$

Step 1

Rewrite $\log_{7} (49 x^{3} \sqrt[5]{y^{2}})$ as $\log_{7} (49 x^{3}) + \log_{7} (\sqrt[5]{y^{2}})$ .

$\log_{7} (49 x^{3}) + \log_{7} (\sqrt[5]{y^{2}})$

Step 2

Rewrite $\log_{7} (49 x^{3})$ as $\log_{7} (49) + \log_{7} (x^{3})$ .

$\log_{7} (49) + \log_{7} (x^{3}) + \log_{7} (\sqrt[5]{y^{2}})$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{y^{2}}$ as $y^{\frac{2}{5}}$ .

$\log_{7} (49) + \log_{7} (x^{3}) + \log_{7} (y^{\frac{2}{5}})$

Step 4

Expand $\log_{7} (x^{3})$ by moving $3$ outside the logarithm.

$\log_{7} (49) + 3 \log_{7} (x) + \log_{7} (y^{\frac{2}{5}})$

Step 5

Expand $\log_{7} (y^{\frac{2}{5}})$ by moving $\frac{2}{5}$ outside the logarithm.

$\log_{7} (49) + 3 \log_{7} (x) + \frac{2}{5} \log_{7} (y)$

Step 6

Logarithm base $7$ of $49$ is $2$ .

$2 + 3 \log_{7} (x) + \frac{2}{5} \log_{7} (y)$

Step 7

Combine $\frac{2}{5}$ and $\log_{7} (y)$ .

$2 + 3 \log_{7} (x) + \frac{2 \log_{7} (y)}{5}$