Precalculus Examples

$\ln (x^{3}) + \ln (\sqrt[4]{y^{3}}) - \ln (1000) + \ln (\sqrt{w})$

Step 1

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

$\ln (x^{3} \sqrt[4]{y^{3}}) - \ln (1000) + \ln (\sqrt{w})$

Step 2

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

$\ln (\frac{x^{3} \sqrt[4]{y^{3}}}{1000}) + \ln (\sqrt{w})$

Step 3

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

$\ln (\frac{x^{3} \sqrt[4]{y^{3}}}{1000} \sqrt{w})$

Step 4

Multiply $\frac{x^{3} \sqrt[4]{y^{3}}}{1000} \sqrt{w}$ .

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Step 4.1

Combine $\frac{x^{3} \sqrt[4]{y^{3}}}{1000}$ and $\sqrt{w}$ .

$\ln (\frac{x^{3} \sqrt[4]{y^{3}} \sqrt{w}}{1000})$

Step 4.2

Rewrite the expression using the least common index of $4$ .

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Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{w}$ as $w^{\frac{1}{2}}$ .

$\ln (\frac{x^{3} (w^{\frac{1}{2}} \sqrt[4]{y^{3}})}{1000})$

Step 4.2.2

Rewrite $w^{\frac{1}{2}}$ as $w^{\frac{2}{4}}$ .

$\ln (\frac{x^{3} (w^{\frac{2}{4}} \sqrt[4]{y^{3}})}{1000})$

Step 4.2.3

Rewrite $w^{\frac{2}{4}}$ as $\sqrt[4]{w^{2}}$ .

$\ln (\frac{x^{3} (\sqrt[4]{w^{2}} \sqrt[4]{y^{3}})}{1000})$

Step 4.3

Combine using the product rule for radicals.

$\ln (\frac{x^{3} \sqrt[4]{w^{2} y^{3}}}{1000})$