Problem 75 - Algebra 2

Question

Simplify each expression. 4 \(\ln e^{2}\)

Step-by-Step Solution

Answer

The simplification result of the expression 4 \(\ln e^{2}\) is 8.

Step 1: Recognize the expression as a natural logarithm

The given expression is 4*\( \ln(e^2) \). The base of the logarithm is e, indicating it is a natural logarithm. The argument of the logarithm is \( e^2 \).

Step 2: Apply the logarithm of a power rule

The logarithm of a power rule states that \( \ln(a^n) = n * \ln(a) \). By applying the rule to \( \ln(e^2) \), it becomes \( 2 * \ln(e) \). The updated expression is 4 * (2 * \( \ln(e) \)).

Step 3: Simplify the expression

We know that \( \ln(e) = 1 \) (since the natural logarithm of e to the base e is 1), hence, the expression simplifies to 4 * (2 * 1). After multiplying, the resulting value is 8.

Key Concepts

Logarithm of a Power Rule

When working with logarithms, a useful tool is the logarithm of a power rule. This rule helps simplify expressions where you encounter an exponent within a logarithmic function. The rule is stated as \( \ln(a^n) = n \times \ln(a) \).

Simply put, when you see a logarithm with an exponent on its argument, you can bring the exponent down to multiply with the logarithm of the base. This simplifies complex logarithmic expressions and makes calculation easier. In our original exercise, the expression \( \ln(e^2) \) simplifies to \( 2 \times \ln(e) \) using this rule.

This approach reduces the complexity of the problem, allowing you to perform further simplifications with ease. Always look out for opportunities to apply this rule when dealing with exponents inside logarithms.

Simplify Expressions

Simplification of expressions is a critical skill in mathematics that makes it easier to solve problems effectively. In the original exercise, we needed to simplify the expression \(4 \times (2 \times \ln(e))\).

In any expression simplification task, follow these basic steps:

• Apply mathematical properties and rules like the commutative or distributive property to combine like terms.
• Perform arithmetic operations step by step, ensuring precision in your calculations.
• Substitute known values, like \( \ln(e) = 1 \), to resolve the expression further.

In the example, using these steps resulted in 4 multiplied by 2, then by 1, thus simplifying to 8. These smaller, more manageable pieces ensure accuracy in your final result.

Properties of Logarithms

Logarithmic properties are an essential part of simplifying expressions when dealing with logarithms. They provide the tools needed to transform and reduce complex expressions into simpler forms.

Three important properties include:

• Product Rule: \( \ln(xy) = \ln(x) + \ln(y) \)
• Quotient Rule: \( \ln\left(\frac{x}{y}\right) = \ln(x) - \ln(y) \)
• Power Rule: (already discussed) \( \ln(a^n) = n \times \ln(a) \)

These rules help break down logarithmic expressions into their components, making them easier to evaluate and understand. For example, in the original exercise, recognizing the power rule helped simplify the logarithmic expression efficiently.

Familiarity with these properties enhances problem-solving capabilities and allows students to confidently tackle math problems involving logarithms.