Precalculus Examples

$\log_{11} (\frac{11 (\sqrt{2})}{3 x})$

Step 1

Rewrite $\log_{11} (\frac{11 (\sqrt{2})}{3 x})$ as $\log_{11} (11 (\sqrt{2})) - \log_{11} (3 x)$ .

$\log_{11} (11 (\sqrt{2})) - \log_{11} (3 x)$

Step 2

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

$\log_{11} (11 \cdot 2^{\frac{1}{2}}) - \log_{11} (3 x)$

Step 3

Rewrite $\log_{11} (11 \cdot 2^{\frac{1}{2}})$ as $\log_{11} (11) + \log_{11} (2^{\frac{1}{2}})$ .

$\log_{11} (11) + \log_{11} (2^{\frac{1}{2}}) - \log_{11} (3 x)$

Step 4

Expand $\log_{11} (2^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{11} (11) + \frac{1}{2} \log_{11} (2) - \log_{11} (3 x)$

Step 5

Logarithm base $11$ of $11$ is $1$ .

$1 + \frac{1}{2} \log_{11} (2) - \log_{11} (3 x)$

Step 6

Combine $\frac{1}{2}$ and $\log_{11} (2)$ .

$1 + \frac{\log_{11} (2)}{2} - \log_{11} (3 x)$

Step 7

Rewrite $\log_{11} (3 x)$ as $\log_{11} (3) + \log_{11} (x)$ .

$1 + \frac{\log_{11} (2)}{2} - (\log_{11} (3) + \log_{11} (x))$

Step 8

Apply the distributive property.

$1 + \frac{\log_{11} (2)}{2} - \log_{11} (3) - \log_{11} (x)$