Precalculus Examples

$\frac{\log_{c} ({(3 x + 1)}^{2} {(3 x - 1)}^{3})}{x^{4}}$

Step 1

Rewrite $\log_{c} ({(3 x + 1)}^{2} {(3 x - 1)}^{3})$ using the change of base formula.

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Step 1.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}{x^{4}}$

Step 1.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Combine.

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

Step 4

Simplify the expression.

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Step 4.1

Multiply $\log ({(3 x + 1)}^{2} {(3 x - 1)}^{3})$ by $1$ .

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

Step 4.2

Reorder factors in $\frac{\log ({(3 x + 1)}^{2} {(3 x - 1)}^{3})}{\log (c) x^{4}}$ .

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