Precalculus Examples

$2 (\ln (x - 5) + \ln (y) - \ln (x + y))$

Step 1

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

$2 (\ln ((x - 5) y) - \ln (x + y))$

Step 2

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

$2 \ln (\frac{(x - 5) y}{x + y})$

Step 3

Simplify $2 \ln (\frac{(x - 5) y}{x + y})$ by moving $2$ inside the logarithm.

$\ln ({(\frac{(x - 5) y}{x + y})}^{2})$

Step 4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1

Apply the product rule to $\frac{(x - 5) y}{x + y}$ .

$\ln (\frac{{((x - 5) y)}^{2}}{{(x + y)}^{2}})$

Step 4.2

Apply the product rule to $(x - 5) y$ .

$\ln (\frac{{(x - 5)}^{2} y^{2}}{{(x + y)}^{2}})$

Step 5

Reorder factors in $\ln (\frac{{(x - 5)}^{2} y^{2}}{{(x + y)}^{2}})$ .

$\ln (\frac{y^{2} {(x - 5)}^{2}}{{(x + y)}^{2}})$