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Chapter 10: Problem 2

What is the base in the expression \(\ln x ?\) A. \(e\) B. 1 C. 10 D. \(x\)

Short Answer

Expert verified
A. \(\e\) is the base of the natural logarithm \(\ln x\).

Step by step solution

01

- Understanding the Natural Logarithm

The notation \(\ln x\) represents the natural logarithm of \(\x\). The natural logarithm is a specific kind of logarithm.
02

- Base of the Natural Logarithm

The natural logarithm \(\ln x\) has a specific base. This base is the mathematical constant \(\e\), which is approximately equal to 2.71828.
03

- Matching with Given Options

From the options provided: A. \(\e\), B. 1, C. 10, and D. \(\x\), we identify that the base \(\e\) corresponds to option A.

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Key Concepts

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

base of logarithm
The base of a logarithm is a crucial concept in understanding how logarithms work. In simple terms, the base of a logarithm is the number that is raised to a given power to get another number. For example, in the expression \(\textrm{log}_b(a)\), \b\ is the base. This tells you the number \b\ must be raised to for you to get \a\.
When we talk about the natural logarithm, commonly denoted as \(\textrm{ln}(x)\), its base is always the mathematical constant \e\, which is approximately equal to 2.71828. Understanding the base is fundamental because it sets the context in which the logarithmic function operates.
mathematical constants
Mathematical constants are numbers with fixed values that arise naturally in mathematics. They are often important in various formulas and equations. One well-known mathematical constant is \(\textbf{e}\).
\e\, also known as Euler's number, is roughly equal to 2.71828.
This number is the base for the natural logarithm and appears in various complex mathematical concepts such as exponential growth and complex numbers.
Another common mathematical constant is \( \pi \approx 3.14159\), often used in calculations involving circles.
Understanding these constants can help simplify complex mathematical problems and are a key part of higher math.
natural logarithm properties
Natural logarithms, indicated as \( \ln(x) \), have unique properties that make them useful in many fields, from calculus to engineering.
Here are a few significant properties:
  • \textbf{Derivative}: The derivative of \( \ln(x) \) with respect to \( \x \) is \( \frac{1}{x} \).

  • \textbf{Integral}: The integral of \( \frac{1}{x} \) is \( \ln|x| + C \), where \( \textbf{C} \) is the constant of integration.

  • \textbf{Inverse}: The natural logarithm \( \ln(x) \) is the inverse function of the base-\textbf{e} exponential function, which means \( \ln(e^x) = x \) and \( \e^{\text{ln}(x)} = x \).

These properties are essential in various calculations and solving exponential equations.

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Most popular questions from this chapter

Find each logarithm. Give approximations to four decimal places. \(\ln \left(7.46 \times e^{3}\right)\)

In your own words, describe the characteristics of the graph of an exponential function. Use the exponential function defined by \(f(x)=3^{x}\) (Exercise 5 ) and the words asymptote, domain, and range in your explanation.

Simplify each expression. Write answers using only positive exponents. See Section 5.1. $$ \frac{5^{-3}}{5^{8}} $$

Concept Check Let \(f(x)=2^{x} .\) We will see in the next section that this function is one-to one. Find each value, always working part (a) before part \((b)\) (a) \(f(4)\) (b) \(f^{-1}(16)\)

Use the change-of-base rule (with either common or natural logarithms) to find logarithm to four decimal places. \(\log _{5} 3\)
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