Precalculus Examples

$\frac{1}{2} \cdot \ln (36) + 2 \ln (4) - \ln (4)$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Simplify $\frac{1}{2} \ln (36)$ by moving $\frac{1}{2}$ inside the logarithm.

$\ln (36^{\frac{1}{2}}) + 2 \ln (4) - \ln (4)$

Step 1.2

Rewrite $36$ as $6^{2}$ .

$\ln ({(6^{2})}^{\frac{1}{2}}) + 2 \ln (4) - \ln (4)$

Step 1.3

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

$\ln (6^{2 (\frac{1}{2})}) + 2 \ln (4) - \ln (4)$

Step 1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.4.1

Cancel the common factor.

$\ln (6^{2 (\frac{1}{2})}) + 2 \ln (4) - \ln (4)$

Step 1.4.2

Rewrite the expression.

$\ln (6^{1}) + 2 \ln (4) - \ln (4)$

Step 1.5

Evaluate the exponent.

$\ln (6) + 2 \ln (4) - \ln (4)$

Step 1.6

Simplify $2 \ln (4)$ by moving $2$ inside the logarithm.

$\ln (6) + \ln (4^{2}) - \ln (4)$

Step 1.7

Raise $4$ to the power of $2$ .

$\ln (6) + \ln (16) - \ln (4)$

Step 2

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

$\ln (6 \cdot 16) - \ln (4)$

Step 3

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

$\ln (\frac{6 \cdot 16}{4})$

Step 4

Cancel the common factor of $6$ and $4$ .

Tap for more steps...

Step 4.1

Factor $2$ out of $6 \cdot 16$ .

$\ln (\frac{2 (3 \cdot 16)}{4})$

Step 4.2

Cancel the common factors.

Tap for more steps...

Step 4.2.1

Factor $2$ out of $4$ .

$\ln (\frac{2 (3 \cdot 16)}{2 (2)})$

Step 4.2.2

Cancel the common factor.

$\ln (\frac{2 (3 \cdot 16)}{2 \cdot 2})$

Step 4.2.3

Rewrite the expression.

$\ln (\frac{3 \cdot 16}{2})$

Step 5

Cancel the common factor of $16$ and $2$ .

Tap for more steps...

Step 5.1

Factor $2$ out of $3 \cdot 16$ .

$\ln (\frac{2 (3 \cdot 8)}{2})$

Step 5.2

Cancel the common factors.

Tap for more steps...

Step 5.2.1

Factor $2$ out of $2$ .

$\ln (\frac{2 (3 \cdot 8)}{2 (1)})$

Step 5.2.2

Cancel the common factor.

$\ln (\frac{2 (3 \cdot 8)}{2 \cdot 1})$

Step 5.2.3

Rewrite the expression.

$\ln (\frac{3 \cdot 8}{1})$

Step 5.2.4

Divide $3 \cdot 8$ by $1$ .

$\ln (3 \cdot 8)$

Step 6

Multiply $3$ by $8$ .

$\ln (24)$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\ln (24)$

Decimal Form:

$3.17805383 \dots$