Precalculus Examples

$3 \log_{5} (3 x + 1) - 2 \log_{5} (2 x - 1) - \log_{5} (x)$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log_{5} (3 x + 1)$ by moving $3$ inside the logarithm.

$\log_{5} ({(3 x + 1)}^{3}) - 2 \log_{5} (2 x - 1) - \log_{5} (x)$

Step 1.2

Simplify $- 2 \log_{5} (2 x - 1)$ by moving $2$ inside the logarithm.

$\log_{5} ({(3 x + 1)}^{3}) - \log_{5} ({(2 x - 1)}^{2}) - \log_{5} (x)$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{5} (\frac{{(3 x + 1)}^{3}}{{(2 x - 1)}^{2}}) - \log_{5} (x)$

Step 3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{5} (\frac{\frac{{(3 x + 1)}^{3}}{{(2 x - 1)}^{2}}}{x})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log_{5} (\frac{{(3 x + 1)}^{3}}{{(2 x - 1)}^{2}} \cdot \frac{1}{x})$

Step 5

Combine.

$\log_{5} (\frac{{(3 x + 1)}^{3} \cdot 1}{{(2 x - 1)}^{2} x})$

Step 6

Simplify the expression.

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Step 6.1

Multiply ${(3 x + 1)}^{3}$ by $1$ .

$\log_{5} (\frac{{(3 x + 1)}^{3}}{{(2 x - 1)}^{2} x})$

Step 6.2

Reorder factors in $\log_{5} (\frac{{(3 x + 1)}^{3}}{{(2 x - 1)}^{2} x})$ .

$\log_{5} (\frac{{(3 x + 1)}^{3}}{x {(2 x - 1)}^{2}})$