Problem 5 - Calculus with Concepts in Calculus

Question

$$ \ln \left(e^{\ln e}\right) $$

Step-by-Step Solution

Answer

1

Step 1: Identify the Innermost Expression

In the given expression \( \ln \left(e^{\ln e} \right) \), begin with the innermost part, which is \( \ln e \). We know that \( \ln e = 1 \) because the natural logarithm of the base \( e \) is 1.

Step 2: Simplify the Exponent

Now replace \( \ln e \) with 1 in the exponent: \( e^{\ln e} = e^1 \). This simplifies our expression to \( e \).

Step 3: Evaluate the Logarithm

Substitute \( e \) back into the logarithm, giving us \( \ln e \). Since \( \ln e = 1 \), the whole expression simplifies to 1.

Step 4: Final Result

The original expression \( \ln \left(e^{\ln e} \right) \) simplifies step-by-step to 1. Thus, the final result of this calculation is 1.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a special type of logarithm where the base is the mathematical constant \( e \). This constant is approximately equal to 2.71828, and it arises naturally in many areas of mathematics, especially in calculus and complex numbers.

Some key properties of the natural logarithm include:

• \( \ln 1 = 0 \) because \( e^0 = 1 \).
• \( \ln e = 1 \) because \( e^1 = e \).
• \( \ln(ab) = \ln a + \ln b \), which means you can break down the logarithm of a product into a sum of logarithms.
• \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \), which allows separation of a logarithm of a fraction into a difference.
• \( \ln(a^b) = b \ln a \), showing how exponents affect the logarithm.
• Since \( e \) is the base, the natural logarithm \( \ln e \) is always 1.

Understanding these rules helps in simplifying expressions that involve natural logarithms. Knowing that \( \ln e = 1 \) is crucial for evaluating the given expression \( \ln(e^{\ln e}) \), which simplifies the problem significantly.

Exponentiation

Exponentiation involves raising a number, known as the base, to a certain power or exponent. In simpler terms, if you have a number \( a \) and you raise it to the power \( b \), you are multiplying \( a \) by itself \( b \) times. Mathematically, this is expressed as \( a^b \).

Important points about exponentiation:

• Any number raised to the power of 0 is 1 (e.g., \( a^0 = 1 \)).
• Raising any number \( a \) to the power of 1 yields \( a \) itself (e.g., \( a^1 = a \)).
• Exponentiation is a powerful operation and is the inverse of taking a logarithm when the base is a natural logarithm.
• The exponential function \( e^x \) is foundational in calculus, describing continuous growth.

In our exercise, understanding that \( e^{\ln e} = e^1 = e \) is crucial for simplifying the expression as it demonstrates the utility of both operations in reversing one another to some extent.

Simplifying Expressions

Simplifying expressions is a process where we make a mathematical expression as straightforward as possible. This often involves reducing the complexity by applying basic arithmetic operations, algebraic identities, and other mathematical rules.

Some steps to effectively simplify an expression:

• Look for the innermost expressions and simplify them first (e.g., \( \ln e = 1 \)).
• Replace exponential terms with simpler equivalents if possible (e.g., \( e^{\ln e} \rightarrow e^1 \)).
• Reapply core properties, such as logarithm identities, to collapse the expression further.
• Check results by reconsidering any potential simplifications (li>Ensure all calculations abide by the mathematical rules and properties.

In our given problem, the steps involved checking the innermost expression \( \ln e = 1 \) first, followed by using properties of exponents to simplify \( e^{1} = e \), and finally reevaluating the logarithmic part to return to a simplified result of 1.