/*! 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 2 What is the base in the expressi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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.

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.

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.

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

Write each logarithmic statement in exponential form. $$ \log _{10} 1=0 $$

Solve each equation. Use natural logarithms. When appropriate, give solutions to three decimal places. See Example 2. $$ \ln e^{0.45 x}=\sqrt{7} $$

Use the change-of-base rule (with either common or natural logarithms) to find logarithm to four decimal places. \(\log _{4} 18\)

According to selected from the last two decades of the 20 th century, the number of trillion cubic feet of dry natural gas consumed worldwide can be approximated by the function defined by $$ f(x)=51.47+6.044 \log _{2} x $$ where \(x=1\) corresponds to \(1980, x=2\) to \(1981,\) and so on. (Source: Energy Information Administration.) Use the function to approximate, to the nearest hundredth, consumption in each year. (a) 1980 (b) 1987 (c) 1995

Consumers can now enjoy movies at home in elaborate home-theater systems. Find the average decibel level \(D=10 \log \left(\frac{I}{I_{0}}\right)\) for each movie with the given intensity \(I\) (a) Avatar; \(5.012 \times 10^{10} \mathrm{I}_{0}\) (b) Iron Man \(2 ; \quad 10^{10} I_{0}\) (c) Clash of the Titans; \(6,310,000,000 \mathrm{I}_{0}\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Statistics

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.