Write as a Single Logarithm 1/3* natural log of 64- natural log of 12

$\frac{1}{3} \cdot \ln (64) - \ln (12)$

Simplify each term.

Simplify $\frac{1}{3} \ln (64)$ by moving $\frac{1}{3}$ inside the logarithm.

$\ln (64^{\frac{1}{3}}) - \ln (12)$

Rewrite $64$ as $4^{3}$ .

$\ln ({(4^{3})}^{\frac{1}{3}}) - \ln (12)$

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

$\ln (4^{3 (\frac{1}{3})}) - \ln (12)$

Cancel the common factor of $3$ .

Cancel the common factor.

$\ln (4^{3 (\frac{1}{3})}) - \ln (12)$

Rewrite the expression.

$\ln (4^{1}) - \ln (12)$

Evaluate the exponent.

$\ln (4) - \ln (12)$

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

$\ln (\frac{4}{12})$

Cancel the common factor of $4$ and $12$ .

Factor $4$ out of $4$ .

$\ln (\frac{4 (1)}{12})$

Cancel the common factors.

Factor $4$ out of $12$ .

$\ln (\frac{4 \cdot 1}{4 \cdot 3})$

Cancel the common factor.

$\ln (\frac{4 \cdot 1}{4 \cdot 3})$

Rewrite the expression.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 1.09861228 \dots$