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 and simplified form of the expression \(\ln(e^{2}/5)\) is \(2 - \ln(5)\)

Step 1: Apply quotient rule

Using the quotient rule, the expression \(\ln(e^{2}/5)\) can be rewritten as \(\ln(e^{2}) - \ln(5)\). The quotient rule works because it states that the logarithm of a quotient can be expressed as a difference of logarithms.

Step 2: Apply transformation rule

Next, apply the transformation rule to simplify \(\ln(e^{2})\). The rule is that \(\ln(e^{n}) = n\), so \(\ln(e^{2}) = 2\). This rule works because the base of natural logarithm is \(e\), and the logarithm of a number to its same base is simply the exponent. Then the expression becomes \(2 - \ln(5)\).

Step 3: Final expression

The logarithm \(\ln(5)\) does not simplify further without a calculator, so the expression stays as is. The final simplified expression is \(2 - \ln(5)\).

Key Concepts

Properties of Logarithms

Logarithms have several properties that make them really useful in simplifying expressions and solving equations. These include:

• Product Rule: States that the logarithm of a product is the sum of the logarithms of the factors. For example, \(\log_b(xy) = \log_b(x) + \log_b(y)\).


• Quotient Rule: This states that the logarithm of a quotient is the difference of the logarithms of the numerator and the denominator. For instance, \(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\).


• Power Rule: Indicates that the logarithm of a power is the exponent times the logarithm of the base. This is, \(\log_b(x^a) = a \cdot \log_b(x)\).


• Change of Base Formula: Allows conversion between different bases, expressed as \(\log_b(x) = \frac{\log_k(x)}{\log_k(b)}\).

By using these properties, complex logarithmic expressions can be rewritten into simpler, equivalent forms. They contribute significantly to reducing the complication involved in evaluating them.

Quotient Rule

The quotient rule simplifies the handling of logarithms when dealing with division inside the log function. The rule essentially transforms a division inside the logarithm into a subtraction outside of it. This rule is written as:

• \(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\)

The beauty of this rule is that it breaks down a more challenging operation into simpler elements—subtraction, which is easier to manage mentally.
To see it in action, consider the example \(\ln(e^{2}/5)\). Using the quotient rule, it is transformed to \(\ln(e^{2}) - \ln(5)\), breaking it down into two simpler parts rather than a division. It's important since it makes the logarithmic expressions more approachable and often aids in further simplification.

Transformation Rule

The transformation rule is particularly useful when the argument of the logarithm is a power of the base. When you encounter a natural logarithm with 'e' as the base, such as \(\ln(e^{n}) \), the transformation rule states that this simplifies directly to the exponent \( n \).

• Expression: \(\ln(e^2) = 2\) because \(\ln(e^n) = n\).

This is because the natural log, by definition, aims to find the power that the base \( e \) must be raised to yield the original argument. Therefore, when the argument is a direct power of \( e \), the calculation is simply the exponent itself.
In our exercise, simplifying \(\ln(e^{2})\) to \(2\) demonstrates the power of the transformation rule. This step essentially eliminates the logarithm, turning it into an arithmetic expression.

Expanding Logarithms

Expanding logarithms involves breaking down complex logarithmic expressions into their simplest components. This often utilizes multiple logarithmic properties to transform the expression into a sum, difference, and possibly products of simpler logarithms. The goal is to make the expression easier to handle or solve.

• From the problem statement: \(\ln\left(\frac{e^{2}}{5}\right)\) is expanded to \(\ln(e^{2}) - \ln(5)\).
• Using the transformation rule on \(\ln(e^{2})\), it simplifies to \(2 - \ln(5)\).

This process of breaking down the expression uses both the quotient rule and transformation rule in a sequence. It shows the versatility and power of logarithmic properties in managing and simplifying expressions. Regular practice with these ways of expanding logarithms sharpens mathematical intuition and problem-solving skills.