Chapter 18: Problem 35
Find the natural logarithm of each number. $$1.845$$
Short Answer
Expert verified
Use a calculator to compute the natural logarithm of 1.845.
Step by step solution
01
Understanding the Task
We are asked to find the natural logarithm of the number 1.845. The natural logarithm is the logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number approximately equal to 2.71828.
02
Using a Calculator
For most numbers, like 1.845, there is no simple method to determine the natural logarithm without the use of a calculator or logarithm table. Enter 1.845 into a calculator and press the ln button (or equivalent depending on the calculator) to find the natural logarithm.
03
Interpreting the Result
The calculator will provide the value of the natural logarithm of 1.845. Record this value as the final answer.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Logarithms
Logarithms are, fundamentally, an operation in mathematics that helps us understand how many times we need to multiply a number (the base) to get another number. For example, when we say \( \log_b a = c \), it means that if we multiply the base \( b \) by itself \( c \) times, we get \( a \).
Specifically, the natural logarithm, often abbreviated as \( \ln \), uses the mathematical constant \( e \) as its base, which makes it a particular kind of logarithm. The natural logarithm of a number \( x \) asks, 'To what power must we raise \( e \) to obtain \( x \)?' Thus, \( \ln(x) \) is the power to which \( e \) must be raised to give the number \( x \). Calculating natural logarithms typically requires the use of a calculator, as they often result in non-integer values with many decimal places.
In simple cases, you might recognize certain values. For instance, \( \ln(1) = 0 \) because \( e^0 = 1 \) and \( \ln(e) = 1 \) because \( e^1 = e \). However, for numbers like 1.845, a calculator is the most practical tool for finding the natural logarithm. Logarithms have wide applications in many fields, including science, engineering, and finance, due to their useful properties and relationship to exponential functions.
Specifically, the natural logarithm, often abbreviated as \( \ln \), uses the mathematical constant \( e \) as its base, which makes it a particular kind of logarithm. The natural logarithm of a number \( x \) asks, 'To what power must we raise \( e \) to obtain \( x \)?' Thus, \( \ln(x) \) is the power to which \( e \) must be raised to give the number \( x \). Calculating natural logarithms typically requires the use of a calculator, as they often result in non-integer values with many decimal places.
In simple cases, you might recognize certain values. For instance, \( \ln(1) = 0 \) because \( e^0 = 1 \) and \( \ln(e) = 1 \) because \( e^1 = e \). However, for numbers like 1.845, a calculator is the most practical tool for finding the natural logarithm. Logarithms have wide applications in many fields, including science, engineering, and finance, due to their useful properties and relationship to exponential functions.
Deciphering the Mathematical Constant e
The mathematical constant \( e \) is a fundamental number in mathematics, just like \( \pi \) (pi). Unlike \( \pi \) which is often recognized in relation to circles, \( e \) is closely associated with growth and decay in calculus and other areas of mathematics dealing with change. Approximately equal to 2.71828, \( e \) is an irrational number, meaning it can't be expressed as a simple fraction and its decimal representation is infinite and non-repeating.
The natural logarithm's base \( e \) arises naturally from compound interest calculations and is the limit of \( (1 + 1/n)^n \) as \( n \) approaches infinity. This makes \( e \) incredibly important in expressing more complex mathematical functions, especially where the rate of change is variable.
The constant \( e \) is also the base of the natural exponential function, \( e^x \), which is its own derivative and integral. This unique property ties together aspects of calculus and analysis, making it a cornerstone for understanding continuous growth or decay.
The natural logarithm's base \( e \) arises naturally from compound interest calculations and is the limit of \( (1 + 1/n)^n \) as \( n \) approaches infinity. This makes \( e \) incredibly important in expressing more complex mathematical functions, especially where the rate of change is variable.
The constant \( e \) is also the base of the natural exponential function, \( e^x \), which is its own derivative and integral. This unique property ties together aspects of calculus and analysis, making it a cornerstone for understanding continuous growth or decay.
Exploring Transcendental Numbers
Transcendental numbers, like \( e \) and \( \pi \) (pi), are numbers that are not algebraic, meaning they are not the solutions to any non-zero polynomial equation with rational coefficients. This distinguishes them from algebraic numbers which can be the solution to such equations. Transcendental numbers are non-repeating, non-terminating decimals that continue infinitely without any pattern.
Their existence is not just a mathematical curiosity; transcendental numbers are crucial in proving certain geometric constructs are impossible, like the famous problem of squaring the circle with only a compass and straightedge. They also show up in many areas of mathematics, from complex analysis to number theory.
The discovery that \( e \) and \( \pi \) are transcendental fundamentally changed the landscape of mathematics, shaping our understanding of the limitations of algebraic solutions and paving the way for further exploration in the realms of analysis and beyond. In the context of logarithms, the transcendental nature of \( e \) ensures that the logarithms of most numbers are unique and non-repeating, which is why a calculator is essential for exact values.
Their existence is not just a mathematical curiosity; transcendental numbers are crucial in proving certain geometric constructs are impossible, like the famous problem of squaring the circle with only a compass and straightedge. They also show up in many areas of mathematics, from complex analysis to number theory.
The discovery that \( e \) and \( \pi \) are transcendental fundamentally changed the landscape of mathematics, shaping our understanding of the limitations of algebraic solutions and paving the way for further exploration in the realms of analysis and beyond. In the context of logarithms, the transcendental nature of \( e \) ensures that the logarithms of most numbers are unique and non-repeating, which is why a calculator is essential for exact values.
