Precalculus Examples

$\log (\frac{10^{x}}{x (x^{2} + 1) (x^{4} + 2)})$

Step 1

Rewrite $\log (\frac{10^{x}}{x (x^{2} + 1) (x^{4} + 2)})$ as $\log (10^{x}) - \log (x (x^{2} + 1) (x^{4} + 2))$ .

$\log (10^{x}) - \log (x (x^{2} + 1) (x^{4} + 2))$

Step 2

Expand $\log (10^{x})$ by moving $x$ outside the logarithm.

$x \log (10) - \log (x (x^{2} + 1) (x^{4} + 2))$

Step 3

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

$x \cdot 1 - \log (x (x^{2} + 1) (x^{4} + 2))$

Step 4

Multiply $x$ by $1$ .

$x - \log (x (x^{2} + 1) (x^{4} + 2))$

Step 5

Rewrite $\log (x (x^{2} + 1) (x^{4} + 2))$ as $\log (x (x^{2} + 1)) + \log (x^{4} + 2)$ .

$x - (\log (x (x^{2} + 1)) + \log (x^{4} + 2))$

Step 6

Rewrite $\log (x (x^{2} + 1))$ as $\log (x) + \log (x^{2} + 1)$ .

$x - (\log (x) + \log (x^{2} + 1) + \log (x^{4} + 2))$

Step 7

Apply the distributive property.

$x - \log (x) - \log (x^{2} + 1) - \log (x^{4} + 2)$