Precalculus Examples

$2 \log (a) - 3 \log (b) + \log (c) - 4 \log (d)$

Step 1

Simplify each term.

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Step 1.1

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

$\log (a^{2}) - 3 \log (b) + \log (c) - 4 \log (d)$

Step 1.2

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

$\log (a^{2}) - \log (b^{3}) + \log (c) - 4 \log (d)$

Step 1.3

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

$\log (a^{2}) - \log (b^{3}) + \log (c) - \log (d^{4})$

Step 2

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

$\log (\frac{a^{2}}{b^{3}}) + \log (c) - \log (d^{4})$

Step 3

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

$\log (\frac{a^{2}}{b^{3}} c) - \log (d^{4})$

Step 4

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

$\log (\frac{\frac{a^{2}}{b^{3}} c}{d^{4}})$

Step 5

Combine $\frac{a^{2}}{b^{3}}$ and $c$ .

$\log (\frac{\frac{a^{2} c}{b^{3}}}{d^{4}})$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{a^{2} c}{b^{3}} \cdot \frac{1}{d^{4}})$

Step 7

Combine.

$\log (\frac{a^{2} c \cdot 1}{b^{3} d^{4}})$

Step 8

Multiply $a^{2}$ by $1$ .

$\log (\frac{a^{2} c}{b^{3} d^{4}})$