Problem 64 - College Algebra and Calculus: An Applied Approach

Question

Use the properties of logarithms to simplify the given logarithmic expression.\(\ln \frac{6}{e^{2}}\)

Step-by-Step Solution

Answer

The simplified form of \( \ln \frac{6}{e^{2}} \) is \( \ln 6 - 2 \).

Step 1: Apply the Quotient Rule

First, apply the quotient rule of logarithms which states: \( \ln \frac{a}{b} = \ln a - \ln b \). Keeping this in mind the expression becomes: \( \ln 6 - \ln e^{2} \).

Step 2: Simplify log of a power

Next, simplify the logarithmic expression of a power \(\ln e^{2} \). According to the properties of logarithms, \(\ln e^{n}\) produces \(n\). Thus our expression becomes: \( \ln 6 - 2 \).

Step 3: Final Simplification

The final expression thus simplifies to: \( \ln 6 - 2 \)

Key Concepts

Quotient Rule

The quotient rule for logarithms is an essential tool for simplifying expressions that involve division inside a logarithm. It states that \[ \ln \left( \frac{a}{b} \right) = \ln a - \ln b \]This rule is quite helpful because it allows us to split complex division inside a logarithm into a simpler subtraction of two separate logs. They become easier to handle individually.
For example, consider the logarithmic expression \( \ln \frac{6}{e^{2}} \). Using the quotient rule, we can break this into \( \ln 6 - \ln e^{2} \). This decomposition is the first step toward simplification.
If you're working with logs and see division inside, remember that the quotient rule is your friend. Use it to turn division into subtraction between separate logarithm terms.

Power Rule

The power rule for logarithms shows us how to handle exponents inside a log. The rule says that the logarithm of a number raised to a power can be simplified by multiplying the power in front of the logarithm:\[ \ln a^{n} = n \times \ln a \]This is fantastic for simplifying expressions where you have powers within logs.
In the expression \( \ln e^{2} \), we apply the power rule. With e being the base of the natural logarithm, the rule simplifies \( \ln e^{2} \) to \( 2 \times \ln e \). Given that \( \ln e = 1 \), it becomes \(2 \times 1\), which is just \(2\).
Applying the power rule reduces the log of any exponential expression down to manageable terms. This step is crucial for simplification, allowing us to transform challenging expressions into straightforward numbers.

Natural Logarithm

Natural logarithms, often written as \( \ln x \), are a specific type of logarithm where the base is the mathematical constant \( e \). This constant is approximately 2.718 and is central to many areas of mathematics.
Natural logarithms are highly useful, especially in fields involving growth and decay processes, such as biology, finance, and physics, because of their unique properties.
Consider this property: \( \ln e = 1 \). This basic fact remains crucial when working with logarithmic expressions involving \( e \). For example, simplifying \( \ln e^{2} \) becomes straightforward once you know that \( \ln e = 1 \). Multiplying the power with \( \ln e \), as done with the power rule, directly gives you the exponent. This highlights how the natural logarithm simplifies computations involving \( e \).