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Magazine Chapter 1: Problem 5
$$ \ln \left(e^{\ln e}\right) $$
Short Answer
Expert verified
1
Step by step solution
01
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.
02
Simplify the Exponent
Now replace \( \ln e \) with 1 in the exponent: \( e^{\ln e} = e^1 \). This simplifies our expression to \( e \).
03
Evaluate the Logarithm
Substitute \( e \) back into the logarithm, giving us \( \ln e \). Since \( \ln e = 1 \), the whole expression simplifies to 1.
04
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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.
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:
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.
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:
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.
