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Chapter 3: Problem 28

Use the properties of logarithms to rewrite and simplify the logarithmic expression. $$\ln \frac{6}{e^{2}}$$

Short Answer

Expert verified
The simplified form of the expression is \(ln(6) - 2\)

Step by step solution

01

using the quotient rule for logarithms

The quotient rule for logarithms states that \(log_b(\frac{m}{n}) = log_b(m) - log_b(n)\). In this case \(b\) is \(e\), \(m\) is 6 and \(n\) is \(e^2\). Hence \(ln(\frac{6}{e^{2}})\) can be rewritten as \(ln(6) - ln(e^{2})\).
02

Applying another property of logarithms

Applying the property of logarithms that states \(log_b(a^n) = n*log_b(a)\), the expression can be simplified further. Since the base for \(ln(e^{2})\) is \(e\), therefore \(ln(e^{2}) = 2*ln(e)\). However, as \(ln(e) = 1\), \(ln(e^2)\) simplifies to \(2\)
03

Final simplification

Substituting \(ln(e^{2})\) by 2 in the result from step 1, we have \(ln(6) - 2\)

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