Problem 13 - Introductory and Intermediate Algebra for College Students 4th

Question

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 simplified form of the log expression is \(2 - \ln(5)\)

Step 1: Express the fraction inside the log as a difference

Recall that the logarithm of a quotient can be expressed as a difference of logarithms. Here, \(\ln \left(\frac{e^{2}}{5}\right)\) can be written as \(\ln(e^{2}) - \ln(5)\)

Step 2: Simplify the Logarithm of \(e^{2}\)

By the property of the natural logarithm, \(\ln(e^{x}) = x\). Therefore, \(\ln(e^{2}) = 2\). So our expression becomes \(2 - \ln(5)\)

Step 3: Final Expression

The final simplified form of the given log expression is \(2 - \ln(5)\). With no further simplification possible, this is the answer.