Problem 14 - College Algebra Essentials

Question

Use properties of logarithms to expand 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

4 - \( \ln{8} \)

Step 1: Applying Division Rule

Using the division rule \( \ln{a/b} = \ln{a} - \ln{b} \), expand the given expression as: \( \ln{e^{4}} - \ln{8} \)

Step 2: Applying Power Rule

Using the power rule \( \ln{a^n} = n\ln{a} \), the first term simplifies to \( 4\ln{e} \). So the expression now becomes \( 4\ln{e} - \ln{8} \)

Step 3: Evaluating Logarithmic Expressions

Since the base of \( \ln{e} \) is \( e \), it equals 1, simplifying the expression to 4 - \( \ln{8} \).

Key Concepts

Understanding the Division Rule of Logarithms

The division rule of logarithms is a fundamental component when working with logarithmic expressions and can greatly help in simplifying complex expressions. The rule states that the logarithm of a division equals the difference of the logarithms. This can be written as:\[\ln\left(\frac{a}{b}\right) = \ln{a} - \ln{b}\]This means that if you have a logarithm involving a fraction or division, you can separate it into two logarithms by subtracting the logarithm of the denominator from the logarithm of the numerator. For instance, for the expression \( \ln\left(\frac{e^{4}}{8}\right) \), we apply this rule to get:

• \( \ln{e^{4}} - \ln{8} \)

Understanding this rule helps in breaking down complex logs into simpler, more manageable parts, allowing us to solve them with ease.

Exploring the Power Rule of Logarithms

The power rule of logarithms is another powerful tool that often accompanies the division rule. It allows us to simplify logs that involve exponents by bringing the exponent outside as a multiplier. The rule is expressed as: \[\ln{a^n} = n \ln{a}\]This makes tackling logarithms with powers a breeze. In our example earlier, notice the term \( \ln{e^{4}} \). Applying the power rule gives us:

• \( 4\ln{e} \)

The exponent 4 is moved in front of the \( \ln{e} \), simplifying the operation. This step not only simplifies but also prepares the expression for further evaluation. This rule is especially useful when involving multiple powers, as it consistently brings order to otherwise complicated terms.

The Tranquility of the Natural Logarithm

The natural logarithm, denoted as \( \ln{x} \), holds a special place in mathematics, particularly because of its base, \( e \), which is approximately equal to 2.718. The natural logarithm has specific properties that simplify calculations substantially, one of which is simplifying \( \ln{e} \). Since it implies \( e \) to the power of itself (essentially \( e^1 \)), \( \ln{e} \) evaluates to 1. Consider the term we obtained from the power rule, \( 4\ln{e} \). Knowing that \( \ln{e} \) is 1 simplifies our expression further to:

• 4 - \( \ln{8} \)

Understanding the natural logarithm's base is crucial for simplifying expressions that may initially seem complicated. Many problems involving \( \ln{x} \) become manageable with this foundational knowledge.

Simplifying Logarithmic Expressions Efficiently

Simplifying complex logarithmic expressions is an essential skill in math, enabling the solving of equations and problems with confidence. Let's look back on the expression we simplified incrementally:

• Start: \( \ln\left(\frac{e^{4}}{8}\right) \)
• Apply division rule: \( \ln{e^{4}} - \ln{8} \)
• Apply power rule: \( 4\ln{e} - \ln{8} \)
• Simplify using \( \ln{e} = 1: 4 - \ln{8} \)

Each step systematically breaks down the expression, turning what might seem complex into something more clear and tractable. Key to this process is recognizing which property to use and accurately transforming the expression according to these properties. Persistently practicing these transformations will make your work with logarithms both quick and precise.