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Chapter 1: Problem 60

Use the properties of logarithms to expand the logarithmic expression. $$ \ln \frac{1}{e} $$

Short Answer

Expert verified
The expanded and simplified expression for \( \ln \frac{1}{e} \) is -1.

Step by step solution

01

Convert Quotient to Difference

First, the logarithm of a quotient can be converted into the difference of two logarithms. In this case, \(\ln \frac{1}{e}\) will become the difference \( \ln (1) - \ln (e) \).
02

Evaluate Logarithms

Next, evaluate each of these individual logarithms. The value of \( \ln (1) \) is 0 because any number raised to the power 0 gives 1. The value of \( \ln (e) \) is 1 because \( e \) raised to the power 1 gives \( e \). In other words, 0 - 1 = -1.

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Key Concepts

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

Properties of Logarithms
Logarithms have several useful properties that make it easier to manipulate and solve equations. Understanding these properties is crucial for expanding and simplifying logarithmic expressions.
  • **Quotient Property:** This property states that the logarithm of a quotient is equal to the difference of the logarithms. Mathematically, this is expressed as \[ \ln \left( \frac{A}{B} \right) = \ln(A) - \ln(B) \]. In the original exercise, we used this property to break down \( \ln \frac{1}{e} \) into \( \ln(1) - \ln(e) \).
  • **Product Property:** This states that the logarithm of a product is the sum of the logarithms. It is given by \[ \ln(AB) = \ln(A) + \ln(B) \]. This is often used when you have multiplication inside the logarithmic expression.
  • **Power Property:** According to this property, the logarithm of a power is equal to the exponent times the logarithm of the base. It can be written as \[ \ln(A^n) = n \cdot \ln(A) \].
These fundamental properties simplify complex logarithmic expressions, making them more manageable.
Logarithmic Expansion
Expanding a logarithmic expression means rewriting it in terms of simpler logarithms. This is particularly helpful for solving equations or simplifying expressions. In the context of the exercise, logarithmic expansion involves breaking down the given expression into more basic parts.
  • In the example exercise, we expanded \( \ln \frac{1}{e} \) using the quotient property. We expressed it as \( \ln(1) - \ln(e) \).
  • Breaking down logarithmic expressions often involves using multiple logarithmic properties. This makes it easier to evaluate or simplify the expressions step by step.
When expanding logarithms, it is crucial to identify which properties will be most helpful. Practicing different problems will build your intuition on which properties to apply, allowing for a seamless expansion of any logarithmic expression.
Natural Logarithm (ln)
The natural logarithm, often denoted as \( \ln(x) \), has a base of Euler's number \( e \), which is approximately 2.718. It is one of the most commonly used logarithms in mathematics, particularly in calculus and complex equations involving exponential growth and decay.
  • The natural logarithm of 1 is always 0. This is because \( e^0 = 1 \). In the provided exercise, we saw that \( \ln(1) = 0 \).
  • The natural logarithm of \( e \) itself is 1. This is because \( e^1 = e \). Therefore, \( \ln(e) = 1 \), a fact used in our exercise to evaluate \( \ln(e) \).
Understanding these basic evaluations can simplify many problems. The natural logarithm finds extensive applications, from calculating growth rates to solving differential equations. Through repeated practice, the evaluations of \( \ln(1) \) and \( \ln(e) \) become intuitive, making it easier to work with natural logarithms in various math problems.

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Most popular questions from this chapter

Describe the difference between a discontinuity that is removable and one that is nonremovable. In your explanation, give examples of the following. (a) A function with a nonremovable discontinuity at \(x=2\) (b) A function with a removable discontinuity at \(x=-2\) (c) A function that has both of the characteristics described in parts (a) and (b)

After an object falls for \(t\) seconds, the speed \(S\) (in feet per second) of the object is recorded in the table. $$ \begin{array}{|l|c|c|c|c|c|c|c|} \hline t & 0 & 5 & 10 & 15 & 20 & 25 & 30 \\ \hline S & 0 & 48.2 & 53.5 & 55.2 & 55.9 & 56.2 & 56.3 \\ \hline \end{array} $$ (a) Create a line graph of the data. (b) Does there appear to be a limiting speed of the object? If there is a limiting speed, identify a possible cause.

Use a graphing utility to graph the given function and the equations \(y=|x|\) and \(y=-|x|\) in the same viewing window. Using the graphs to visually observe the Squeeze Theorem, find \(\lim _{x \rightarrow 0} f(x)\). $$ h(x)=x \cos \frac{1}{x} $$

Prove that if \(\lim _{x \rightarrow c} f(x)\) exists and \(\lim _{x \rightarrow c}[f(x)+g(x)]\) does not exist, then \(\lim _{x \rightarrow c} g(x)\) does not exist.

Writing Use a graphing utility to graph \(f(x)=x, \quad g(x)=\sin x, \quad\) and \(\quad h(x)=\frac{\sin x}{x}\) in the same viewing window. Compare the magnitudes of \(f(x)\) and \(g(x)\) when \(x\) is "close to" \(0 .\) Use the comparison to write \(a\) short paragraph explaining why \(\lim _{x \rightarrow 0} h(x)=1\).
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