Write as a Single Logarithm 1/3( natural log of 27+2* natural log of 8)

$\frac{1}{3} (\ln (27) + 2 \cdot \ln (8))$

Step 1

Simplify each term.

Simplify $2 \ln (8)$ by moving $2$ inside the logarithm.

$\frac{1}{3} \cdot (\ln (27) + \ln (8^{2}))$

Raise $8$ to the power of $2$ .

$\frac{1}{3} \cdot (\ln (27) + \ln (64))$

Step 2

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

$\frac{1}{3} \cdot \ln (27 \cdot 64)$

Step 3

Multiply $27$ by $64$ .

$\frac{1}{3} \cdot \ln (1728)$

Step 4

Simplify $\frac{1}{3} \ln (1728)$ by moving $\frac{1}{3}$ inside the logarithm.

$\ln (1728^{\frac{1}{3}})$

Step 5

Simplify the expression.

Rewrite $1728$ as $12^{3}$ .

$\ln ({(12^{3})}^{\frac{1}{3}})$

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

$\ln (12^{3 (\frac{1}{3})})$

Step 6

Cancel the common factor of $3$ .

Cancel the common factor.

$\ln (12^{3 (\frac{1}{3})})$

Rewrite the expression.

$\ln (12^{1})$

Step 7

Evaluate the exponent.

$\ln (12)$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\ln (12)$

Decimal Form:

$2.48490664 \dots$