Calculus Examples

$\log (x {(x^{5} + 8)}^{- \frac{1}{2}})$

Step 1

Rewrite $\log (x {(x^{5} + 8)}^{- \frac{1}{2}})$ as $\log (x) + \log ({(x^{5} + 8)}^{- \frac{1}{2}})$ .

$\log (x) + \log ({(x^{5} + 8)}^{- \frac{1}{2}})$

Step 2

Expand $\log ({(x^{5} + 8)}^{- \frac{1}{2}})$ by moving $- \frac{1}{2}$ outside the logarithm.

$\log (x) - \frac{1}{2} \log (x^{5} + 8)$

Step 3

Combine $\log (x^{5} + 8)$ and $\frac{1}{2}$ .

$\log (x) - \frac{\log (x^{5} + 8)}{2}$

Step 4

To write $\log (x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\log (x) \cdot \frac{2}{2} - \frac{\log (x^{5} + 8)}{2}$

Step 5

Combine $\log (x)$ and $\frac{2}{2}$ .

$\frac{\log (x) \cdot 2}{2} - \frac{\log (x^{5} + 8)}{2}$

Step 6

Combine the numerators over the common denominator.

$\frac{\log (x) \cdot 2 - \log (x^{5} + 8)}{2}$

Step 7

Move $2$ to the left of $\log (x)$ .

$\frac{2 \log (x) - \log (x^{5} + 8)}{2}$