Problem 42 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms to rewrite and simplify the logarithmic expression.$$\ln 8 e^{3}$$.

Step-by-Step Solution

Answer

The simplified form of the expression \( \ln 8 e^{3} \) is 5.07944.

Step 1: Break down the logarithmic expression using logarithm properties

The first step would be to apply the property of logarithms that says the logarithm of a product is the sum of the logarithms. Therefore, the given expression \( \ln 8 e^{3} \) can be rewritten as \( \ln 8 + \ln e^{3} \).

Step 2: Simplify \( \ln e^{3} \)

The natural logarithm of \( e^{n} \) is \( n \). Therefore, \( \ln e^{3} \) can be simplified to 3. So, our simplified expression becomes \( \ln 8 + 3 \).

Step 3: Evaluate \( \ln 8 \)

The natural logarithm \( \ln 8 \) can be approximated to 2.07944. So, the final simplified expression becomes \( 2.07944 + 3 \).

Step 4: Summarize the final expression

Adding 2.07944 and 3 results in 5.07944. This is the simplified form of the given logarithmic expression.

Key Concepts

Natural Logarithm

Understanding the natural logarithm is crucial for simplifying logarithmic expressions. The natural logarithm, denoted as \( \text{ln}(x) \), is a logarithm with base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. It often appears in mathematics and the physical sciences because of its unique properties related to growth rates and decay processes.

In the context of the original exercise, recognizing that \( \text{ln}(e^3) = 3 \) is an application of the fundamental property of natural logarithms that \( \text{ln}(e^n) = n \). This relationship is pivotal for correctly simplifying expressions that involve \( e \), the base of the natural logarithm.

Logarithm Properties

Several key logarithm properties enable the simplification of logarithmic expressions. One primary property is that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Formally, this property means that \( \text{log}_b(mn) = \text{log}_b(m) + \text{log}_b(n) \), where \( b \) is the base of the logarithms involved.

Logarithm of a Power

Another property shown in the exercise is the logarithm of a power, which states that \( \text{log}_b(m^n) = n * \text{log}_b(m) \). However, when dealing with a natural logarithm, the logarithm of \( e \) to any power simplifies directly to the exponent: \( \text{ln}(e^n) = n \).

Logarithm of \( e \)

It is also indispensable to remember that the natural logarithm of \( e \) itself, \( \text{ln}(e) \), is always 1. This is because the base and the argument are the same, satisfying the property \( \text{log}_b(b) = 1 \).

Simplifying Logarithmic Expressions

The exercise provides a framework for simplifying complex logarithmic expressions using the aforementioned properties. Simplification often involves breaking down expressions into smaller parts that can be evaluated more easily and then combining those results to find the simplified expression.

To simplify effectively, always look for opportunities to split a logarithm of a product into a sum of logarithms, or to reduce a logarithm of a power to a multiple of the logarithm of the base. Following the step-by-step solution provided, start by applying the product property and then utilize any simplifications that natural logarithms allow. Finally, round out the process by performing the necessary arithmetic operations as seen in the final step of the exercise.

Teaching advice includes practicing these techniques with a variety of expressions to build familiarity and ease in recognizing these properties in different contexts. Combining practice with the memorization of core properties results in a robust understanding of how to simplify logarithmic expressions.