Precalculus Examples

$\ln (2) - \ln (y) + \frac{1}{2} \cdot \ln (x + y) - \frac{1}{3} \cdot \ln (27)$

Step 1

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

$\ln (\frac{2}{y}) + \frac{1}{2} \cdot \ln (x + y) - \frac{1}{3} \cdot \ln (27)$

Step 2

Simplify each term.

Tap for more steps...

Step 2.1

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

$\ln (\frac{2}{y}) + \ln ({(x + y)}^{\frac{1}{2}}) - \frac{1}{3} \cdot \ln (27)$

Step 2.2

Multiply $- \frac{1}{3} \ln (27)$ .

Tap for more steps...

Step 2.2.1

Reorder $\ln (27)$ and $\frac{1}{3}$ .

$\ln (\frac{2}{y}) + \ln ({(x + y)}^{\frac{1}{2}}) - (\frac{1}{3} \ln (27))$

Step 2.2.2

Simplify $\frac{1}{3} \ln (27)$ by moving $\frac{1}{3}$ inside the logarithm.

$\ln (\frac{2}{y}) + \ln ({(x + y)}^{\frac{1}{2}}) - \ln (27^{\frac{1}{3}})$

Step 2.3

Rewrite $27$ as $3^{3}$ .

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

Step 2.4

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

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

Step 2.5

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.5.1

Cancel the common factor.

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

Step 2.5.2

Rewrite the expression.

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

Step 2.6

Evaluate the exponent.

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

Step 3

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

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

Step 4

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

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

Step 5

Combine $\frac{2}{y}$ and ${(x + y)}^{\frac{1}{2}}$ .

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

Step 6

Multiply the numerator by the reciprocal of the denominator.

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

Step 7

Combine.

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

Step 8

Simplify the expression.

Tap for more steps...

Step 8.1

Multiply $2$ by $1$ .

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

Step 8.2

Move $3$ to the left of $y$ .

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