/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 9 Simplify the following expressio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 9

Simplify the following expressions. \(e^{e^{\ln 1}}\)

Short Answer

Expert verified
The simplified expression is \(e\).

Step by step solution

01

Simplify the Natural Logarithm

First, identify the innermost expression, which is the natural logarithm of 1: \(\ln 1\). Recall that the natural logarithm function returns the exponent to which the base e must be raised to produce the argument. Since any number raised to the zero power is 1, we have \(\ln 1 = 0\).
02

Substitute and Simplify

Replace the \(\ln 1\) in the original expression with 0. So the expression \(e^{e^{\ln 1}}\) becomes \(e^{e^0}\).
03

Simplify the Exponential

Now, simplify the exponent part. Anything raised to the power of 0 is 1, hence \(e^0 = 1\). Therefore, the expression reduces to \(e^1\).
04

Final Simplification

Since \(e^1\) is just e, the simplified expression is \(e\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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 fundamental mathematical function. It answers the question: To what power must e (approximately 2.718) be raised to get a particular number? For instance, \(\ln 1\) is asking, 'e raised to which power equals 1?' Since \(\text{e}^0 = 1\), \(\text{ln} 1 = 0\). This is the key idea used in the exercise to simplify the given expression. Understanding the natural logarithm is crucial in calculus and various scientific fields.

Key Properties:
  • ewline The base of the natural logarithm is e.
  • The natural logarithm of e, \(\text{ln} e = 1\).
  • \(\text{ln}(ab) = \text{ln}(a) + \text{ln}(b)\): This is the property of logarithms, called the product rule.
exponential function
The exponential function, typically written as \(e^x\), is a critical function in mathematics. It grows rapidly and is the inverse function of the natural logarithm. One of the most essential properties of the exponential function is \(e^0 = 1\). This was applied in the exercise when simplifying the expression \(e^{e^0}\) because \(e^0 = 1\). The exponential function models many natural phenomena, such as population growth and radioactive decay.

Key Characteristics:
  • The base is always e (approximately 2.718).
  • The slope of the exponential function's graph increases exponentially.
  • ewline It has an asymptote at y=0, meaning the function approaches zero but never actually reaches it.
mathematical simplification
Mathematical simplification involves reducing a complex expression into its simplest form. This technique makes calculations easier and expressions more readable. In the exercise, we first identified the innermost expression (the natural logarithm of 1), simplified it to 0, and then used properties of exponents to further reduce the expression step by step. Simplification requires a solid understanding of mathematical rules and properties.

Steps in Simplification:
  • Identify and simplify inner expressions first.
  • Use mathematical properties such as \(e^0 = 1\) and \(e^1 = e\).
  • Break down complex expressions into simpler, more manageable steps.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Differentiate the following functions. \(y=\ln \left(e^{x}+e^{-x}\right)\)

Solve the following equations for \(x .\) \(\left(e^{2}\right)^{x} \cdot e^{\ln 1}=4\)

Evaluate the given expression. Use \(\ln 2=.69\) and \(\ln 3=1.1\). (a) \(\ln 12\) (b) \(\ln 16\) (c) \(\ln \left(9 \cdot 2^{4}\right)\)

Differentiate. \(y=(\ln 4 x)(\ln 2 x)\)

Differentiate. \(y=\ln \left[\sqrt{x e^{x^{2}+1}}\right]\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.