Problem 73 - Algebra 2

Question

Simplify each expression. 2 \(\ln e\)

Step-by-Step Solution

Answer

The simplified expression of \(2 \ln e\) is 2.

Step 1: Recognize the Property of Natural Logarithm

This is a very straightforward step. We remember that natural logarithm of e is equal to 1, which can be written as \(\ln e = 1\). This fact directly stems from the definition of the natural logarithm function.

Step 2: Apply the Property to the Given Expression

Now we substitute this fact into the original problem. Original problem is \(2 \ln e\). Applying the information from step 1, we can replace \(\ln e\) with 1. So, the problem now becomes: \(2 * 1\).

Step 3: Compute the Result

Finally, 2 * 1 = 2. So, the simplified expression of \(2 \ln e\) becomes 2.

Key Concepts

Logarithm Properties

Logarithms have specific properties that simplify various mathematical expressions. Understanding these properties is crucial. One fundamental property of the natural logarithm is its relationship with Euler's number, denoted as \( e \). The natural logarithm of \( e \) is equal to 1, represented as \( \ln e = 1 \). This means that when you take the natural logarithm of \( e \), the result is always 1.

There are several key properties of logarithms that can help simplify expressions efficiently:

• Product Property: \( \ln(a \cdot b) = \ln a + \ln b \).
• Quotient Property: \( \ln \left(\frac{a}{b}\right) = \ln a - \ln b \).
• Power Property: \( \ln(a^b) = b \cdot \ln a \).

Understanding these properties allows you to break down complex logarithmic expressions into simpler parts. In our exercise, knowing that \( \ln e = 1 \) allowed us to easily transform \( 2 \ln e \) to \( 2 \cdot 1 \). This step uses the fundamental property of \( e \), showcasing the power of logarithmic properties in simplifying calculations.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra and calculus. It involves using mathematical properties to transform the expression into its simplest form. In the exercise, we started with the expression \( 2 \ln e \).

To simplify this expression, we use the known property \( \ln e = 1 \). By substituting \( \ln e \) with 1, the expression simplifies from \( 2 \ln e \) to \( 2 \times 1 \). This substitution step makes our job easier, as we move from a logarithmic expression to a simple multiplication.

Here are some steps usually helpful in the simplification task:

• Identify known values or properties that could substitute parts of the expression.
• Use basic arithmetic operations to calculate further.
• Always check if the expression can be simplified further by applying other mathematical rules.

In our exercise, substituting and calculating was enough to reach the simplest form of '2', showing the power of substitution in simplifying expressions.

Mathematical Computations

Performing mathematical computations involves applying arithmetic rules to get precise solutions. It can include addition, subtraction, multiplication, and division. In our exercise, after substituting \( \ln e = 1 \) into \(2 \ln e\), we arrived at \( 2 \times 1 \).

The multiplication step here is straightforward. You multiply 2 by 1, which remains as 2.

Understanding this simple operation solidly establishes the foundation for tackling more complex problems. Mathematical computations can be made easier by following specific strategies:

• Break down the expression into elementary calculations.
• Use substitution wherever possible to work with simpler numbers or equations.
• Double-check each step to ensure accuracy in computations.

By mastering these basic computations, more complex algebraic or calculus-related problems become much easier to solve.