Question

Use the properties of logarithms to expand the logarithmic expression. $$ \ln \left(3 e^{2}\right) $$

Short answer

\(\ln(3e^2)\) expands to \(\ln(3) + 2\).

Step-by-step solution

Apply Logarithm Product Rule

Using the logarithm product rule, \( \ln(ab) = \ln(a) + \ln(b) \), we express the logarithm as sum of two logarithms. Here, a = 3 and b = $e^2$, so we get: \( \ln(3e^2) = \ln(3) + \ln(e^2) \).

Apply Power Rule

The power rule indicates that logarithm of a number raised to an exponent is equal to the exponent times the logarithm of the number, i.e., \( \ln(a^b) = b \ln(a) \). Hence for the second term \( \ln(e^2) \), we move the exponent, 2, in front of ln(e): \( \ln(e^2) = 2 \ln(e) \).

Simplify Logarithm of e

The logarithm base e of e to the power of any integer n equals the integer, i.e., \( \ln(e^n) = n \). In this situation, \( \ln(e) = 1 \) so we can simplify \( 2 \ln(e) \) to 2.

Substitute back into Equation

Substitute Steps 2 and 3 back into the equation obtained in Step 1: \( \ln(3e^2) = \ln(3) + 2 \).