Calculus Examples

$\frac{1}{2} \ln (8) - \frac{1}{2} \ln (2)$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Simplify $\frac{1}{2} \ln (8)$ by moving $\frac{1}{2}$ inside the logarithm.

$\ln (8^{\frac{1}{2}}) - \frac{1}{2} \cdot \ln (2)$

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

$\ln (8^{\frac{1}{2}}) - (\frac{1}{2} \ln (2))$

Step 1.2.2

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

$\ln (8^{\frac{1}{2}}) - \ln (2^{\frac{1}{2}})$

Step 2

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

$\ln (\frac{8^{\frac{1}{2}}}{2^{\frac{1}{2}}})$

Step 3

Use the power of quotient rule $\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m}$ .

$\ln ({(\frac{8}{2})}^{\frac{1}{2}})$

Step 4

Simplify the expression.

Tap for more steps...

Step 4.1

Divide $8$ by $2$ .

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

Step 4.2

Rewrite $4$ as $2^{2}$ .

$\ln ({(2^{2})}^{\frac{1}{2}})$

Step 4.3

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

$\ln (2^{2 (\frac{1}{2})})$

Step 5

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.1

Cancel the common factor.

$\ln (2^{2 (\frac{1}{2})})$

Step 5.2

Rewrite the expression.

$\ln (2^{1})$

Step 6

Evaluate the exponent.

$\ln (2)$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\ln (2)$

Decimal Form:

$0.69314718 \dots$