Problem 13 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \ln \left(\frac{e^{2}}{5}\right) $$

Step-by-Step Solution

Answer

The expanded form of the given expression \(\ln \left(\frac{e^{2}}{5}\right)\) is \(2 - \ln(5)\).

Step 1: Apply Quotient Rule

According to the rule of logarithm, the division inside the log can be changed into subtraction outside of the log. Therefore, \(\ln \left(\frac{e^{2}}{5}\right)\) will become \(\ln(e^2) - \ln(5)\)

Step 2: Apply the Power Rule

A power inside of a log can be moved out front and turned into a multiplier, according to the power rule of logarithms. Considering this, \(\ln(e^2)\) can be rewritten as \(2 \cdot \ln(e)\)

Step 3: Simplify the Logarithmic Expression

The value of \(\ln(e)\) is 1. Substituting this into the expression from the previous step, we get \(2 \cdot 1\) or 2. So the expanded expression is \(2 - \ln(5)\)