Problem 54 - College Algebra and Calculus: An Applied Approach

Question

Find the exact value of the logarithmic expression without using a calculator.\(\ln \sqrt[4]{e^{3}}\)

Step-by-Step Solution

Answer

The exact value of the logarithmic expression is \(3/4\).

Step 1: Recognize 'e' as the base of natural logarithms

Since the base of the natural logarithm, \(\ln\), is \(e\), it means that \(\ln e = 1\)

Step 2: Interpret the expression \(\sqrt[4]{e^{3}}\)

This expression denotes the fourth root of \(e^{3}\), which can also be written as \((e^{3})^{1/4}\). According to the law of exponents, this equals to \(e^{3(1/4)} = e^{3/4}\)

Step 3: Apply the properties of the logarithm

Now we find \(\ln \sqrt[4]{e^{3}} = \ln e^{3/4}\). Since the exponent in a logarithmic expression can be brought in front as a multiplier, the expression can be simplified to \(3/4 * \ln e\)

Step 4: Simplify the expression

Now remember that \(\ln e = 1\). Therefore, the given expression can be simplified as \(3/4 * 1 = 3/4\)

Key Concepts

Properties of Logarithms

Logarithms are a powerful tool in mathematics, especially when it comes to simplifying expressions. When working with logarithms, it's essential to understand their fundamental properties. These properties allow us to manipulate and simplify logarithmic expressions effectively.
Here are three essential properties to know:

• **Product Property**: The logarithm of a product can be expressed as the sum of the logarithms: \( \log_b (xy) = \log_b x + \log_b y \).
• **Quotient Property**: The logarithm of a quotient can be represented as the difference of the logarithms: \( \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \).
• **Power Property**: The logarithm of a power can be simplified by bringing the exponent in front as a multiplier: \( \log_b (x^a) = a \cdot \log_b x \).

In this particular problem, we use the power property to simplify the expression \( \ln e^{3/4} \) by bringing the exponent \( 3/4 \) in front, resulting in \( 3/4 \cdot \ln e \).
Remembering these properties can significantly simplify the process of solving logarithmic expressions, making them more manageable and straightforward.

Exponentiation

Exponentiation is a mathematical operation involving numbers or expressions that are raised to a certain power. In simpler terms, it means multiplying a number by itself a specified number of times.
When dealing with roots, exponentiation provides a convenient way to express them. For example, taking the fourth root of a number is the same as raising it to the power of \( \frac{1}{4} \).
In the expression \( \sqrt[4]{e^3} \), the fourth root is represented using exponentiation as \((e^3)^{1/4}\), which simplifies to \(e^{3/4}\).
Understanding exponentiation is crucial in mathematics. It not only helps with simplifying complex expressions but also allows you to approach problems like this exercise with confidence.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves using various properties and mathematical rules to make expressions easier to work with. The goal is to reach a reduced form that is simpler to understand and calculate.
In this exercise, simplifying the expression \( \ln \sqrt[4]{e^3} \) involves several steps. First, interpret the nested root using exponents, resulting in \( \ln e^{3/4} \). Next, apply the power property of logarithms to bring the exponent down, turning it into \( 3/4 \cdot \ln e \).
Since \( \ln e \) equals 1, this expression further simplifies to \( 3/4 \). By executing these steps, you effectively turn a seemingly complicated expression into a simple fraction.
Mastering these techniques not only aids in solving textbook problems but also enhances your overall mathematical confidence.