Calculus Examples

$\ln (u^{2} + 1) + \ln (\sin (u)) - \frac{1}{2} \cdot \ln (u v)$

Step 1

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

$\ln ((u^{2} + 1) \sin (u)) - \frac{1}{2} \cdot \ln (u v)$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$\ln (u^{2} \sin (u) + 1 \sin (u)) - \frac{1}{2} \cdot \ln (u v)$

Step 2.2

Multiply $\sin (u)$ by $1$ .

$\ln (u^{2} \sin (u) + \sin (u)) - \frac{1}{2} \cdot \ln (u v)$

Step 2.3

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

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Step 2.3.1

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

$\ln (u^{2} \sin (u) + \sin (u)) - (\frac{1}{2} \ln (u v))$

Step 2.3.2

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

$\ln (u^{2} \sin (u) + \sin (u)) - \ln ({(u v)}^{\frac{1}{2}})$

Step 2.4

Apply the product rule to $u v$ .

$\ln (u^{2} \sin (u) + \sin (u)) - \ln (u^{\frac{1}{2}} v^{\frac{1}{2}})$

Step 3

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

$\ln (\frac{u^{2} \sin (u) + \sin (u)}{u^{\frac{1}{2}} v^{\frac{1}{2}}})$

Step 4

Factor $\sin (u)$ out of $u^{2} \sin (u) + \sin (u)$ .

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Step 4.1

Factor $\sin (u)$ out of $u^{2} \sin (u)$ .

$\ln (\frac{\sin (u) u^{2} + \sin (u)}{u^{\frac{1}{2}} v^{\frac{1}{2}}})$

Step 4.2

Multiply by $1$ .

$\ln (\frac{\sin (u) u^{2} + \sin (u) \cdot 1}{u^{\frac{1}{2}} v^{\frac{1}{2}}})$

Step 4.3

Factor $\sin (u)$ out of $\sin (u) u^{2} + \sin (u) \cdot 1$ .

$\ln (\frac{\sin (u) (u^{2} + 1)}{u^{\frac{1}{2}} v^{\frac{1}{2}}})$