Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \ln (3)$ by moving $2$ inside the logarithm.

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

Step 1.2

Raise $3$ to the power of $2$ .

$\ln (9) + \frac{1}{3} \cdot \ln (x) + \ln ({(e)}^{y}) - \frac{1}{2} \cdot \ln (z)$

Step 1.3

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

$\ln (9) + \ln (x^{\frac{1}{3}}) + \ln ({(e)}^{y}) - \frac{1}{2} \cdot \ln (z)$

Step 1.4

Use logarithm rules to move $y$ out of the exponent.

$\ln (9) + \ln (x^{\frac{1}{3}}) + y \ln (e) - \frac{1}{2} \cdot \ln (z)$

Step 1.5

The natural logarithm of $e$ is $1$ .

$\ln (9) + \ln (x^{\frac{1}{3}}) + y \cdot 1 - \frac{1}{2} \cdot \ln (z)$

Step 1.6

Multiply $y$ by $1$ .

$\ln (9) + \ln (x^{\frac{1}{3}}) + y - \frac{1}{2} \cdot \ln (z)$

Step 1.7

Multiply $- \frac{1}{2} \ln (z)$ .

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Step 1.7.1

Reorder $\ln (z)$ and $\frac{1}{2}$ .

$\ln (9) + \ln (x^{\frac{1}{3}}) + y - (\frac{1}{2} \ln (z))$

Step 1.7.2

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

$\ln (9) + \ln (x^{\frac{1}{3}}) + y - \ln (z^{\frac{1}{2}})$

Step 2

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

$\ln (9 x^{\frac{1}{3}}) + y - \ln (z^{\frac{1}{2}})$

Step 3

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

$y + \ln (\frac{9 x^{\frac{1}{3}}}{z^{\frac{1}{2}}})$