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Chapter 12: Problem 60

Evaluate each expression without using a calculator. $$\ln e$$

Short Answer

Expert verified
The natural logarithm of e, \(ln(e)\), is 1.

Step by step solution

01

Understanding the natural logarithm

The natural logarithm, denoted as ln, is the inverse of the exponential function with base e. Put simply, if \(y = e^x\), then the natural logarithm of y is x, or \(ln(y) = x\). That is, the exponent to which e must be raised to obtain a number is called the natural logarithm of the number.
02

Applying the definition

By the definition of the natural logarithm set out in the previous step, the natural logarithm of e is the power to which we must raise e to obtain e. Since \(e^1 = e\), \(ln(e) = 1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Function
Dive into the world of exponential functions, a crucial concept in mathematics. An exponential function is a mathematical expression in the form of \( f(x) = e^x \), where \( e \) is the base of the natural logarithm鈥攁 mathematical constant approximately equal to 2.71828. These functions are called exponential because the variable \( x \) is in the exponent.

Exponential functions are powerful tools. They are used to model growth processes like populations or investments.
  • A key property of the exponential function is its rapid rate of increase鈥攊t grows much faster than polynomial functions.
  • Graphically, exponential functions have a distinctive curve that starts slowly and then rises sharply.
Recognizing this function is essential for understanding the concept of natural logarithms, which are their inverses.
Inverse Functions
Inverse functions are like mathematical mirrors. They reverse the effect of another function. For example, if a function takes \( a \) to \( b \), its inverse will take \( b \) back to \( a \).

The natural logarithm \( \ln(x) \) is the inverse of the exponential function \( e^x \). This means that if \( y = e^x \), then \( x = \ln(y) \). Understanding inverse functions is essential since they help us "undo" operations.
  • The notation for an inverse function often uses \( f^{-1}(x) \), but for logarithms, we specifically use \( \ln(x) \).
  • Graphically, the inverse function reflects the original function over the line \( y = x \).
Knowing how to work with inverse functions allows you to solve equations relating exponential growth and decay.
Mathematical Constants
Mathematical constants are numbers with a special significance in math. They are universally recognized and play critical roles in various formulas and theorems. One such constant is \( e \), which is the base of the natural logarithm.

\( e \) is not just any number; it has unique properties that make it indispensable in calculus, particularly in continuous growth and decay processes.
  • The value of \( e \) is approximately 2.71828, but it is an irrational number, meaning it cannot be exactly expressed as a simple fraction.
  • \( e \) is found in many areas of mathematics such as compound interest, Euler's identity, and even in probability theory.
Understanding \( e \) with its fascinating attributes, helps demystify many mathematical equations and applications.

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Most popular questions from this chapter

Solve the system: $$\left\\{\begin{aligned}2 x &=11-5 y \\\3 x-2 y &=-12\end{aligned}\right.$$ (Section 4.3, Example 4)

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I can use any positive number other than 1 in the changeof-base property, but the only practical bases are 10 and \(e\) because my calculator gives logarithms for these two bases.

Complete the table for a savings account subject to continuous compounding ( \(A=P e^{n}\) ). Round answers to one decimal place. $$\begin{array}{l|c|l|c} \hline \begin{array}{l} \text { Amount } \\ \text { Invested } \end{array} & \begin{array}{l} \text { Annual Interest } \\ \text { Rate } \end{array} & \begin{array}{l} \text { Accumulated } \\ \text { Amount } \end{array} & \begin{array}{l} \text { Time } t \\ \text { in Years } \end{array} \\ \hline \$ 17,425 & 4.25 \% & \$ 25,000 & \\ \hline \end{array}$$

The function \(P(t)=145 e^{-0.092 t}\) models a runner's pulse, \(P(t),\) in beats per minute, \(t\) minutes after a race, where \(0 \leq t \leq 15 .\) Graph the function using a graphing utility. [TRACE] along the graph and determine after how many minutes the runner's pulse will be 70 beats per minute. Round to the nearest tenth of a minute. Verify your observation algebraically.

a. Simplify: \(e^{\ln 3}\) b. Use your simplification from part (a) to rewrite \(3^{x}\) in terms of base \(e\)
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