Calculus Examples

$\frac{1}{2} \cdot \ln (x y) + \frac{3}{2} \cdot \ln (\frac{x}{y})$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{2} \ln (x y)$ by moving $\frac{1}{2}$ inside the logarithm.

$\ln ({(x y)}^{\frac{1}{2}}) + \frac{3}{2} \cdot \ln (\frac{x}{y})$

Step 1.2

Apply the product rule to $x y$ .

$\ln (x^{\frac{1}{2}} y^{\frac{1}{2}}) + \frac{3}{2} \cdot \ln (\frac{x}{y})$

Step 1.3

Simplify $\frac{3}{2} \ln (\frac{x}{y})$ by moving $\frac{3}{2}$ inside the logarithm.

$\ln (x^{\frac{1}{2}} y^{\frac{1}{2}}) + \ln ({(\frac{x}{y})}^{\frac{3}{2}})$

Step 1.4

Apply the product rule to $\frac{x}{y}$ .

$\ln (x^{\frac{1}{2}} y^{\frac{1}{2}}) + \ln (\frac{x^{\frac{3}{2}}}{y^{\frac{3}{2}}})$

Step 2

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

$\ln (x^{\frac{1}{2}} y^{\frac{1}{2}} \frac{x^{\frac{3}{2}}}{y^{\frac{3}{2}}})$

Step 3

Cancel the common factor of $y^{\frac{1}{2}}$ .

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Step 3.1

Factor $y^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} y^{\frac{1}{2}}$ .

$\ln (y^{\frac{1}{2}} x^{\frac{1}{2}} \frac{x^{\frac{3}{2}}}{y^{\frac{3}{2}}})$

Step 3.2

Factor $y^{\frac{1}{2}}$ out of $y^{\frac{3}{2}}$ .

$\ln (y^{\frac{1}{2}} x^{\frac{1}{2}} \frac{x^{\frac{3}{2}}}{y^{\frac{1}{2}} y^{\frac{2}{2}}})$

Step 3.3

Cancel the common factor.

$\ln (y^{\frac{1}{2}} x^{\frac{1}{2}} \frac{x^{\frac{3}{2}}}{y^{\frac{1}{2}} y^{\frac{2}{2}}})$

Step 3.4

Rewrite the expression.

$\ln (x^{\frac{1}{2}} \frac{x^{\frac{3}{2}}}{y^{\frac{2}{2}}})$

Step 4

Combine $x^{\frac{1}{2}}$ and $\frac{x^{\frac{3}{2}}}{y^{\frac{2}{2}}}$ .

$\ln (\frac{x^{\frac{1}{2}} x^{\frac{3}{2}}}{y^{\frac{2}{2}}})$

Step 5

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{3}{2}}$ by adding the exponents.

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Step 5.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\ln (\frac{x^{\frac{1}{2} + \frac{3}{2}}}{y^{\frac{2}{2}}})$

Step 5.2

Combine the numerators over the common denominator.

$\ln (\frac{x^{\frac{1 + 3}{2}}}{y^{\frac{2}{2}}})$

Step 5.3

Add $1$ and $3$ .

$\ln (\frac{x^{\frac{4}{2}}}{y^{\frac{2}{2}}})$

Step 5.4

Divide $4$ by $2$ .

$\ln (\frac{x^{2}}{y^{\frac{2}{2}}})$

Step 6

Cancel the common factor of $2$ .

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Step 6.1

Cancel the common factor.

$\ln (\frac{x^{2}}{y^{\frac{2}{2}}})$

Step 6.2

Rewrite the expression.

$\ln (\frac{x^{2}}{y^{1}})$

Step 7

Simplify.

$\ln (\frac{x^{2}}{y})$