Calculus Examples

$\log (10 x \sqrt{x - 3})$

Step 1

Rewrite $\log (10 x \sqrt{x - 3})$ as $\log (10 x) + \log (\sqrt{x - 3})$ .

$\log (10 x) + \log (\sqrt{x - 3})$

Step 2

Rewrite $\log (10 x)$ as $\log (10) + \log (x)$ .

$\log (10) + \log (x) + \log (\sqrt{x - 3})$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 3}$ as ${(x - 3)}^{\frac{1}{2}}$ .

$\log (10) + \log (x) + \log ({(x - 3)}^{\frac{1}{2}})$

Step 4

Expand $\log ({(x - 3)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log (10) + \log (x) + \frac{1}{2} \log (x - 3)$

Step 5

Logarithm base $10$ of $10$ is $1$ .

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

Step 6

Combine $\frac{1}{2}$ and $\log (x - 3)$ .

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

Step 7

Write $1$ as a fraction with a common denominator.

$\frac{2}{2} + \frac{\log (x - 3)}{2} + \log (x)$

Step 8

Combine the numerators over the common denominator.

$\frac{2 + \log (x - 3)}{2} + \log (x)$