Problem 55 - Precalculus : Building Concepts and Connections

Question

In Exercises \(53-64,\) simplify each expression. Assume that each variable expression is defined for appropriate values of \(x .\) Do not use a calculator. $$\ln e^{\sqrt{3}}$$

Step-by-Step Solution

Answer

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

Step 1: Identify the given expression

The given expression is \(\ln e^{\sqrt{3}}\). This is the natural logarithm of e raised to the square root of 3.

Step 2: Use the exponential-logarithmic identity

According to the exponential-logarithmic identity, whenever the base of the natural logarithm (ln) and the base of the exponent are the same, the result is equal to the exponent. So, \(\ln e^{\sqrt{3}} = \sqrt{3}\).

Key Concepts

Exponential Logarithmic Identity

The exponential-logarithmic identity is a fundamental principle in mathematics that helps simplify expressions involving logarithms and exponents. To understand this identity, it's important to have a grasp of the natural logarithm and the base of the natural logarithm, which is the constant \(e\) (approximately 2.718). This identity is based on the idea that when you take the natural logarithm of \(e\) raised to any power, the result simplifies to just the exponent.
Consider the expression \(\ln e^{\sqrt{3}}\). To simplify it using the exponential-logarithmic identity, we note that the base of the natural logarithm \(\ln\) is \(e\). Likewise, the base of the exponent here is also \(e\). So when these two bases are the same, the natural logarithm cancels out the exponentiation, and we are left with the exponent value. In this case, the expression simplifies to \(\sqrt{3}\).
This identity helps mathematicians move quickly between exponential and logarithmic forms of expressions. It removes complexity and reveals underlying relationships between the numbers.

Simplifying Expressions

Simplifying expressions is a crucial skill in mathematics. It allows you to transform a complex expression into a simpler and more manageable form. This process often involves identifying patterns or utilizing certain mathematical identities and properties.
In our exercise, we started with \(\ln e^{\sqrt{3}}\). By recognizing the exponential-logarithmic identity, we could directly simplify it to \(\sqrt{3}\).
Several strategies are commonly used in simplifying expressions:

• Identifying common patterns or identities, like the exponential-logarithmic identity.
• Combining like terms, which involves merging terms that have the same variables raised to the same power.
• Applying distributive, associative, or commutative properties to reorganize or break down expressions.

For students, mastering these techniques in simplifying expressions can make problem-solving much more straightforward, especially in exams where time efficiency is important.

Mathematics Education

Understanding complex concepts like logarithms and exponents is a key part of mathematics education. These concepts appear in various branches of mathematics, from algebra to calculus, and are essential for students planning to study advanced mathematics or related fields.
Mathematics education often focuses on teaching students to solve problems through step-by-step processes. This approach helps students learn not only how to reach the correct answer but also understand the reasoning behind it.
However, to improve understanding further, students should:

• Engage in regular practice to become comfortable with these identities and concepts.
• Use resources like textbooks, online platforms, or tutorials that provide clear explanations and numerous examples.
• Seek help from teachers or peers when a concept is confusing. Collaborative learning often clarifies difficult ideas.

By fostering a deeper understanding of principles like the exponential-logarithmic identity, students can build strong problem-solving skills, which are vital in both academic and real-world scenarios.