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

Add $2$ and $1$ .

$x \cdot 1 - \log (x \cdot x^{3} (x^{4} + 2))$

Step 5

Multiply $x$ by $1$ .

$x - \log (x \cdot x^{3} (x^{4} + 2))$

Step 6

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 6.1

Multiply $x$ by $x^{3}$ .

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Step 6.1.1

Raise $x$ to the power of $1$ .

$x - \log (x^{1} x^{3} (x^{4} + 2))$

Step 6.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x - \log (x^{1 + 3} (x^{4} + 2))$

Step 6.2

Add $1$ and $3$ .

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

Step 7

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

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

Step 8

Expand $\log (x^{4})$ by moving $4$ outside the logarithm.

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

Step 9

Simplify each term.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Multiply $4$ by $- 1$ .

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