Precalculus Examples

$4 \ln (u) - \frac{1}{2} \cdot \ln (w^{2})$

Step 1

Simplify each term.

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Step 1.1

Simplify $4 \ln (u)$ by moving $4$ inside the logarithm.

$\ln (u^{4}) - \frac{1}{2} \cdot \ln (w^{2})$

Step 1.2

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

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Step 1.2.1

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

$\ln (u^{4}) - (\frac{1}{2} \ln (w^{2}))$

Step 1.2.2

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

$\ln (u^{4}) - \ln ({(w^{2})}^{\frac{1}{2}})$

Step 1.3

Multiply the exponents in ${(w^{2})}^{\frac{1}{2}}$ .

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Step 1.3.1

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

$\ln (u^{4}) - \ln (w^{2 (\frac{1}{2})})$

Step 1.3.2

Cancel the common factor of $2$ .

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Step 1.3.2.1

Cancel the common factor.

$\ln (u^{4}) - \ln (w^{2 (\frac{1}{2})})$

Step 1.3.2.2

Rewrite the expression.

$\ln (u^{4}) - \ln (w^{1})$

Step 1.4

Simplify.

$\ln (u^{4}) - \ln (w)$

Step 2

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

$\ln (\frac{u^{4}}{w})$