Precalculus Examples

$\log_{5} (\frac{7 \sqrt{u}}{u^{3}})$

Step 1

Rewrite $\log_{5} (\frac{7 \sqrt{u}}{u^{3}})$ as $\log_{5} (7 \sqrt{u}) - \log_{5} (u^{3})$ .

$\log_{5} (7 \sqrt{u}) - \log_{5} (u^{3})$

Step 2

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

$\log_{5} (7 u^{\frac{1}{2}}) - \log_{5} (u^{3})$

Step 3

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

$\log_{5} (7 u^{\frac{1}{2}}) - (3 \log_{5} (u))$

Step 4

Multiply $3$ by $- 1$ .

$\log_{5} (7 u^{\frac{1}{2}}) - 3 \log_{5} (u)$

Step 5

Rewrite $\log_{5} (7 u^{\frac{1}{2}})$ as $\log_{5} (7) + \log_{5} (u^{\frac{1}{2}})$ .

$\log_{5} (7) + \log_{5} (u^{\frac{1}{2}}) - 3 \log_{5} (u)$

Step 6

Expand $\log_{5} (u^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{5} (7) + \frac{1}{2} \log_{5} (u) - 3 \log_{5} (u)$

Step 7

Combine $\frac{1}{2}$ and $\log_{5} (u)$ .

$\log_{5} (7) + \frac{\log_{5} (u)}{2} - 3 \log_{5} (u)$

Step 8

To write $- 3 \log_{5} (u)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\log_{5} (7) + \frac{\log_{5} (u)}{2} - 3 \log_{5} (u) \cdot \frac{2}{2}$

Step 9

Combine $- 3 \log_{5} (u)$ and $\frac{2}{2}$ .

$\log_{5} (7) + \frac{\log_{5} (u)}{2} + \frac{- 3 \log_{5} (u) \cdot 2}{2}$

Step 10

Combine the numerators over the common denominator.

$\log_{5} (7) + \frac{\log_{5} (u) - 3 \log_{5} (u) \cdot 2}{2}$

Step 11

Simplify each term.

Tap for more steps...

Step 11.1

Simplify the numerator.

Tap for more steps...

Step 11.1.1

Factor $\log_{5} (u)$ out of $\log_{5} (u) - 3 \log_{5} (u) \cdot 2$ .

Tap for more steps...

Step 11.1.1.1

Multiply by $1$ .

$\log_{5} (7) + \frac{\log_{5} (u) \cdot 1 - 3 \log_{5} (u) \cdot 2}{2}$

Step 11.1.1.2

Factor $\log_{5} (u)$ out of $- 3 \log_{5} (u) \cdot 2$ .

$\log_{5} (7) + \frac{\log_{5} (u) \cdot 1 + \log_{5} (u) (- 3 \cdot 2)}{2}$

Step 11.1.1.3

Factor $\log_{5} (u)$ out of $\log_{5} (u) \cdot 1 + \log_{5} (u) (- 3 \cdot 2)$ .

$\log_{5} (7) + \frac{\log_{5} (u) (1 - 3 \cdot 2)}{2}$

Step 11.1.2

Multiply $- 3$ by $2$ .

$\log_{5} (7) + \frac{\log_{5} (u) (1 - 6)}{2}$

Step 11.1.3

Subtract $6$ from $1$ .

$\log_{5} (7) + \frac{\log_{5} (u) \cdot - 5}{2}$

Step 11.2

Move $- 5$ to the left of $\log_{5} (u)$ .

$\log_{5} (7) + \frac{- 5 \cdot \log_{5} (u)}{2}$

Step 11.3

Move the negative in front of the fraction.

$\log_{5} (7) - \frac{5 \log_{5} (u)}{2}$