Problem 14 - 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. $$\ln \left(\frac{e^{4}}{8}\right)$$

Step-by-Step Solution

Answer

The expanded form of the expression \(\ln \left(\frac{e^{4}}{8}\right)\) is 4 − \ln(8).

Step 1: Identify the parts of the logarithmic expression

Firstly, identify that you're given a natural log function (\ln) of a fraction. This fraction can be split up into two separate logarithms through the rule: \ln(a / b) = \ln(a) - \ln(b). The fraction is \(\frac{e^4}{8}\), so \(a = e^4\) and \(b = 8\).\n

Step 2: Apply the Logarithm Division Rule

Next, use the logarithm division rule mentioned above to split the function into two parts: \ln(e^4)-\ln(8). Now your expression is simplified and can be executed separately.\n

Step 3: Simplify \ln(e^4)

The power rule for logarithms can be expressed as \ln(a^n)= n* \ln(a). Thus, \ln(e^4)= 4* \ln(e). As the logarithm base \(e\) of \(e\) is 1, therefore \ln(e^4)=4.\n

Step 4: Evaluate \ln(8)

The value \ln(8) cannot be simplified further without a calculator because 8 is not a power of \(e\). Thus, we leave it as is.\n

Step 5: Combine the results

Now we combine the results from Step 3 and Step 4. The final expanded form of the original expression is now 4 - \ln(8).\n