Precalculus Examples

$5 \ln (x) - 2 \ln (x) - 7 \ln (z) + 21 \ln (w)$

Step 1

Simplify each term.

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Step 1.1

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

$\ln (x^{5}) - 2 \ln (x) - 7 \ln (z) + 21 \ln (w)$

Step 1.2

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

$\ln (x^{5}) - \ln (x^{2}) - 7 \ln (z) + 21 \ln (w)$

Step 1.3

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

$\ln (x^{5}) - \ln (x^{2}) - \ln (z^{7}) + 21 \ln (w)$

Step 1.4

Simplify $21 \ln (w)$ by moving $21$ inside the logarithm.

$\ln (x^{5}) - \ln (x^{2}) - \ln (z^{7}) + \ln (w^{21})$

Step 2

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

$\ln (\frac{x^{5}}{x^{2}}) - \ln (z^{7}) + \ln (w^{21})$

Step 3

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

$\ln (\frac{\frac{x^{5}}{x^{2}}}{z^{7}}) + \ln (w^{21})$

Step 4

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

$\ln (\frac{\frac{x^{5}}{x^{2}}}{z^{7}} w^{21})$

Step 5

Cancel the common factor of $x^{5}$ and $x^{2}$ .

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Step 5.1

Factor $x^{2}$ out of $x^{5}$ .

$\ln (\frac{\frac{x^{2} x^{3}}{x^{2}}}{z^{7}} w^{21})$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Multiply by $1$ .

$\ln (\frac{\frac{x^{2} x^{3}}{x^{2} \cdot 1}}{z^{7}} w^{21})$

Step 5.2.2

Cancel the common factor.

$\ln (\frac{\frac{x^{2} x^{3}}{x^{2} \cdot 1}}{z^{7}} w^{21})$

Step 5.2.3

Rewrite the expression.

$\ln (\frac{\frac{x^{3}}{1}}{z^{7}} w^{21})$

Step 5.2.4

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

$\ln (\frac{x^{3}}{z^{7}} w^{21})$

Step 6

Combine $\frac{x^{3}}{z^{7}}$ and $w^{21}$ .

$\ln (\frac{x^{3} w^{21}}{z^{7}})$