Write as a Single Logarithm 3 natural log of 10-4 natural log of 2- natural log of 5

$3 \ln (10) - 4 \ln (2) - \ln (5)$

Simplify each term.

Simplify $3 \ln (10)$ by moving $3$ inside the logarithm.

$\ln (10^{3}) - 4 \ln (2) - \ln (5)$

Raise $10$ to the power of $3$ .

$\ln (1000) - 4 \ln (2) - \ln (5)$

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

$\ln (1000) - \ln (2^{4}) - \ln (5)$

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

$\ln (1000) - \ln (16) - \ln (5)$

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

$\ln (\frac{1000}{16}) - \ln (5)$

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

$\ln (\frac{\frac{1000}{16}}{5})$

Cancel the common factor of $1000$ and $16$ .

Factor $8$ out of $1000$ .

$\ln (\frac{\frac{8 (125)}{16}}{5})$

Cancel the common factors.

Factor $8$ out of $16$ .

$\ln (\frac{\frac{8 \cdot 125}{8 \cdot 2}}{5})$

Cancel the common factor.

$\ln (\frac{\frac{8 \cdot 125}{8 \cdot 2}}{5})$

Rewrite the expression.

$\ln (\frac{\frac{125}{2}}{5})$

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{125}{2} \cdot \frac{1}{5})$

Cancel the common factor of $5$ .

Factor $5$ out of $125$ .

$\ln (\frac{5 (25)}{2} \cdot \frac{1}{5})$

Cancel the common factor.

$\ln (\frac{5 \cdot 25}{2} \cdot \frac{1}{5})$

Rewrite the expression.

$\ln (\frac{25}{2})$

The result can be shown in multiple forms.

Exact Form:

$\ln (\frac{25}{2})$

Decimal Form:

$2.52572864 \dots$