Calculus Examples

$\frac{1}{2} \log (x) - 3 \log (\sin (2 x)) + 2$

Step 1

Simplify each term.

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Step 1.1

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

$\log (x^{\frac{1}{2}}) - 3 \log (\sin (2 x)) + 2$

Step 1.2

Simplify $- 3 \log (\sin (2 x))$ by moving $3$ inside the logarithm.

$\log (x^{\frac{1}{2}}) - \log (\sin^{3} (2 x)) + 2$

Step 2

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

$\log (\frac{x^{\frac{1}{2}}}{\sin^{3} (2 x)}) + 2$

Step 3

Simplify each term.

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Step 3.1

Multiply by $1$ .

$\log (\frac{x^{\frac{1}{2}}}{\sin^{3} (2 x) \cdot 1}) + 2$

Step 3.2

Separate fractions.

$\log (\frac{x^{\frac{1}{2}}}{1} \cdot \frac{1}{\sin^{3} (2 x)}) + 2$

Step 3.3

Convert from $\frac{1}{\sin^{3} (2 x)}$ to $\csc^{3} (2 x)$ .

$\log (\frac{x^{\frac{1}{2}}}{1} \csc^{3} (2 x)) + 2$

Step 3.4

Divide $x^{\frac{1}{2}}$ by $1$ .

$\log (x^{\frac{1}{2}} \csc^{3} (2 x)) + 2$