Precalculus Examples

$3 \log (x) - 2 \log (4) - 4 \log (y) + 2 \log (3)$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log (x)$ by moving $3$ inside the logarithm.

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

Step 1.2

Simplify $- 2 \log (4)$ by moving $2$ inside the logarithm.

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

Step 1.3

Raise $4$ to the power of $2$ .

$\log (x^{3}) - \log (16) - 4 \log (y) + 2 \log (3)$

Step 1.4

Simplify $- 4 \log (y)$ by moving $4$ inside the logarithm.

$\log (x^{3}) - \log (16) - \log (y^{4}) + 2 \log (3)$

Step 1.5

Simplify $2 \log (3)$ by moving $2$ inside the logarithm.

$\log (x^{3}) - \log (16) - \log (y^{4}) + \log (3^{2})$

Step 1.6

Raise $3$ to the power of $2$ .

$\log (x^{3}) - \log (16) - \log (y^{4}) + \log (9)$

Step 2

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

$\log (\frac{x^{3}}{16}) - \log (y^{4}) + \log (9)$

Step 3

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

$\log (\frac{\frac{x^{3}}{16}}{y^{4}}) + \log (9)$

Step 4

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (\frac{\frac{x^{3}}{16}}{y^{4}} \cdot 9)$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{x^{3}}{16} \cdot \frac{1}{y^{4}} \cdot 9)$

Step 6

Combine.

$\log (\frac{x^{3} \cdot 1}{16 y^{4}} \cdot 9)$

Step 7

Multiply $x^{3}$ by $1$ .

$\log (\frac{x^{3}}{16 y^{4}} \cdot 9)$

Step 8

Combine $\frac{x^{3}}{16 y^{4}}$ and $9$ .

$\log (\frac{x^{3} \cdot 9}{16 y^{4}})$

Step 9

Move $9$ to the left of $x^{3}$ .

$\log (\frac{9 x^{3}}{16 y^{4}})$