Question

expand the expression and simplify. \\(\ln(5e^2)\\). do not give a decimal value. (20 points)
\\(\square + \ln(\square)\\)

Explanation:

Step1: Use logarithm product rule

The logarithm product rule states that $\ln(ab) = \ln(a) + \ln(b)$ for $a > 0$ and $b > 0$. For the expression $\ln(5e^{2})$, we can apply this rule where $a = 5$ and $b = e^{2}$. So, $\ln(5e^{2})=\ln(5)+\ln(e^{2})$.

Step2: Simplify $\ln(e^{2})$

We know the property of logarithms that $\ln(e^{x})=x$ (since the natural logarithm $\ln$ and the exponential function $e^{x}$ are inverse functions). Here, $x = 2$, so $\ln(e^{2}) = 2$.

Step3: Rewrite the expression

Substituting the simplified form of $\ln(e^{2})$ back into the expression from Step 1, we get $\ln(5e^{2})=2+\ln(5)$.

Answer:

$2$ + $\ln(\boldsymbol{5})$