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Chapter 3: Problem 97

Evaluate or simplify each expression without using a calculator. $$e^{\ln 5 x^{2}}$$

Short Answer

Expert verified
The simplified form of the expression \( e^{ln 5x^{2}} \) is \( 5x^{2} \)

Step by step solution

01

Identify the expression

The given expression is \( e^{ln 5x^{2}} \)
02

Apply the property of Natural logarithm and exponential function

The base of the natural logarithm \( ln \) is the Euler's number \( e \), and they cancel each other out. So, we apply this property and simplify the expression. The expression becomes \( 5x^{2} \) which is the inside of the exponent.
03

Rewrite the simplified expression

Following from step 2, the simplified expression is simply \( 5x^{2} \)

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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, often denoted as \( \ln(x) \), is a special type of logarithm where the base is Euler's number \( e \), which is approximately 2.718. The natural logarithm function is the inverse of the exponential function with base \( e \).

In simpler terms, if you have \( y = e^x \), then \( x = \ln(y) \). The beauty of this relationship lies in their ability to 'cancel' each other out in equations.
  • When you see an expression like \( e^{\ln(a)} \), it simplifies directly to \( a \).
  • The \( e \) and \( \ln \) perform opposite actions. Thus, in our exercise, \( e^{\ln(5x^2)} \) simplifies directly to \( 5x^2 \).


Understanding this principle makes solving these equations straightforward. You can recognize the symmetry between \( e \) and \( \ln \), and use it to simplify complex expressions effectively.
Euler's Number
Euler's number, known as \( e \), is a fundamental constant in mathematics. Approximated as 2.718, \( e \) is unique because it is the base of the natural logarithm and naturally occurs in various growth processes.

Beyond just logarithms, \( e \) is significant in areas like calculus, due to properties such as the derivative of \( e^x \) being itself \( e^x \).
  • \( e \) holds a special property: \( e^{\ln(x)} = x \), showcasing its role in connecting exponential and logarithmic functions.
  • \( e \) is central in continuously compounded interest and population growth models.


By understanding \( e \) and its properties, you unlock a deeper comprehension of many natural processes and mathematical concepts, making it an essential part of the learning journey.
Simplification of Expressions
The process of simplifying expressions involves making them easier to read or solve. It often relies on understanding mathematical properties and rules.

In our exercise, the expression \( e^{\ln(5x^2)} \) reduces to \( 5x^2 \) seamlessly through simplification.
  • Use the inverse relationship of \( e \) and \( \ln \) to simplify expressions quickly, remembering that \( e^{\ln(a)} = a \).
  • This approach efficiently reduces complex logarithmic and exponential expressions, masking them more manageable.


This skill is not only crucial for accurate calculations, but it also aids in understanding advanced topics in algebra, calculus, and beyond. Mastery in simplification strategies saves time and enhances problem-solving capabilities.

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Most popular questions from this chapter

Without using a calculator, determine which is the greater number: \(\log _{4} 60\) or \(\log _{3} 40\).

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$$\text { Solve for } y: 7 x+3 y=18$$

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