Simplify/Condense 2 natural log of 3+5 natural log of 2- natural log of 8

$2 \ln (3) + 5 \ln (2) - \ln (8)$

Step 1

Simplify each term.

Simplify $2 \ln (3)$ by moving $2$ inside the logarithm.

$\ln (3^{2}) + 5 \ln (2) - \ln (8)$

Raise $3$ to the power of $2$ .

$\ln (9) + 5 \ln (2) - \ln (8)$

Simplify $5 \ln (2)$ by moving $5$ inside the logarithm.

$\ln (9) + \ln (2^{5}) - \ln (8)$

Raise $2$ to the power of $5$ .

$\ln (9) + \ln (32) - \ln (8)$

Step 2

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

$\ln (9 \cdot 32) - \ln (8)$

Step 3

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

$\ln (\frac{9 \cdot 32}{8})$

Step 4

Cancel the common factor of $32$ and $8$ .

Factor $8$ out of $9 \cdot 32$ .

$\ln (\frac{8 (9 \cdot 4)}{8})$

Cancel the common factors.

Factor $8$ out of $8$ .

$\ln (\frac{8 (9 \cdot 4)}{8 (1)})$

Cancel the common factor.

$\ln (\frac{8 (9 \cdot 4)}{8 \cdot 1})$

Rewrite the expression.

$\ln (\frac{9 \cdot 4}{1})$

Divide $9 \cdot 4$ by $1$ .

$\ln (9 \cdot 4)$

Step 5

Multiply $9$ by $4$ .

$\ln (36)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\ln (36)$

Decimal Form:

$3.58351893 \dots$