Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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