Precalculus Examples

$20 \ln (7) - 2 \ln (12) - 3 \ln (x) + 5 \ln (y)$

Step 1

Simplify each term.

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Step 1.1

Simplify $20 \ln (7)$ by moving $20$ inside the logarithm.

$\ln (7^{20}) - 2 \ln (12) - 3 \ln (x) + 5 \ln (y)$

Step 1.2

Raise $7$ to the power of $20$ .

$\ln (79792266297612000) - 2 \ln (12) - 3 \ln (x) + 5 \ln (y)$

Step 1.3

Simplify $- 2 \ln (12)$ by moving $2$ inside the logarithm.

$\ln (79792266297612000) - \ln (12^{2}) - 3 \ln (x) + 5 \ln (y)$

Step 1.4

Raise $12$ to the power of $2$ .

$\ln (79792266297612000) - \ln (144) - 3 \ln (x) + 5 \ln (y)$

Step 1.5

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

$\ln (79792266297612000) - \ln (144) - \ln (x^{3}) + 5 \ln (y)$

Step 1.6

Simplify $5 \ln (y)$ by moving $5$ inside the logarithm.

$\ln (79792266297612000) - \ln (144) - \ln (x^{3}) + \ln (y^{5})$

Step 2

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

$\ln (\frac{79792266297612000}{144}) - \ln (x^{3}) + \ln (y^{5})$

Step 3

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

$\ln (\frac{\frac{79792266297612000}{144}}{x^{3}}) + \ln (y^{5})$

Step 4

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

$\ln (\frac{\frac{79792266297612000}{144}}{x^{3}} y^{5})$

Step 5

Cancel the common factor of $79792266297612000$ and $144$ .

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Step 5.1

Factor $48$ out of $79792266297612000$ .

$\ln (\frac{\frac{48 (1662338881200250)}{144}}{x^{3}} y^{5})$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $48$ out of $144$ .

$\ln (\frac{\frac{48 \cdot 1662338881200250}{48 \cdot 3}}{x^{3}} y^{5})$

Step 5.2.2

Cancel the common factor.

$\ln (\frac{\frac{48 \cdot 1662338881200250}{48 \cdot 3}}{x^{3}} y^{5})$

Step 5.2.3

Rewrite the expression.

$\ln (\frac{\frac{1662338881200250}{3}}{x^{3}} y^{5})$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{1662338881200250}{3} \cdot \frac{1}{x^{3}} y^{5})$

Step 7

Combine.

$\ln (\frac{1662338881200250 \cdot 1}{3 x^{3}} y^{5})$

Step 8

Multiply $1662338881200250$ by $1$ .

$\ln (\frac{1662338881200250}{3 x^{3}} y^{5})$

Step 9

Combine $\frac{1662338881200250}{3 x^{3}}$ and $y^{5}$ .

$\ln (\frac{1662338881200250 y^{5}}{3 x^{3}})$