Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Simplify $- 2 \ln (x^{2 + y^{2}})$ by moving $2$ inside the logarithm.

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

Step 1.3

Multiply the exponents in ${(x^{2 + y^{2}})}^{2}$ .

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Step 1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Multiply $2$ by $2$ .

$\ln ({(x - y)}^{\frac{3}{2}}) - \ln (x^{4 + y^{2} \cdot 2})$

Step 1.3.4

Move $2$ to the left of $y^{2}$ .

$\ln ({(x - y)}^{\frac{3}{2}}) - \ln (x^{4 + 2 y^{2}})$

Step 2

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

$\ln (\frac{{(x - y)}^{\frac{3}{2}}}{x^{4 + 2 y^{2}}})$