Precalculus Examples

$5 \ln (x) - \frac{1}{2} \cdot \ln (y) + 6 \ln (z)$

Step 1

Simplify each term.

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Step 1.1

Simplify $5 \ln (x)$ by moving $5$ inside the logarithm.

$\ln (x^{5}) - \frac{1}{2} \cdot \ln (y) + 6 \ln (z)$

Step 1.2

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

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Step 1.2.1

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

$\ln (x^{5}) - (\frac{1}{2} \ln (y)) + 6 \ln (z)$

Step 1.2.2

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

$\ln (x^{5}) - \ln (y^{\frac{1}{2}}) + 6 \ln (z)$

Step 1.3

Simplify $6 \ln (z)$ by moving $6$ inside the logarithm.

$\ln (x^{5}) - \ln (y^{\frac{1}{2}}) + \ln (z^{6})$

Step 2

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

$\ln (\frac{x^{5}}{y^{\frac{1}{2}}}) + \ln (z^{6})$

Step 3

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

$\ln (\frac{x^{5}}{y^{\frac{1}{2}}} z^{6})$

Step 4

Combine $\frac{x^{5}}{y^{\frac{1}{2}}}$ and $z^{6}$ .

$\ln (\frac{x^{5} z^{6}}{y^{\frac{1}{2}}})$