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Chapter 3: Problem 40

Given \(\ln 4=1.3863\) and \(\ln 5=1.6094,\) find each value. Do not use a calculator. $$ \ln \frac{1}{4} $$

Short Answer

Expert verified
\(\ln \frac{1}{4} = -1.3863\)

Step by step solution

01

Understanding the property

We need to recall the logarithmic property that states \( \ln \left(\frac{1}{x}\right) = -\ln x\). This tells us that the natural logarithm of the reciprocal of a number is the opposite of the natural logarithm of the number itself.
02

Applying the property to the problem

Given \(\ln 4 = 1.3863\), we can use the property \(\ln \left(\frac{1}{x}\right) = -\ln x\) to find \(\ln \frac{1}{4}\):\[\ln \frac{1}{4} = -\ln 4 = -1.3863\]

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Key Concepts

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

Natural Logarithm
The natural logarithm, typically denoted as "ln," is a special kind of logarithm that uses the mathematical constant \( e \) as its base. \( e \), approximately equal to 2.71828, is an irrational number like \( \pi \) and is important in mathematics, particularly in calculus and complex analysis. Here's why natural logarithms are useful:
  • They simplify the process of dealing with exponential growth or decay problems, which occur frequently in nature.
  • They help in solving equations involving exponentials since it converts these into linear form.
To express the natural logarithm of a number \( x \), we write \( \ln x \). For example, the given values \( \ln 4 = 1.3863 \) and \( \ln 5 = 1.6094 \) tell us that \( e^{1.3863} \approx 4 \) and \( e^{1.6094} \approx 5 \). The natural logarithm, therefore, helps us discover how many times we must multiply \( e \) to get a certain number.
Reciprocal Rule
The Reciprocal Rule is a key concept when working with logarithms, especially natural logarithms. It is expressed with the formula:\[\ln \left( \frac{1}{x} \right) = -\ln x\]This rule tells us that the natural logarithm of the reciprocal of a number is the negative of the natural logarithm of the number itself. Here's a step-by-step breakdown:
  • The reciprocal of a number \( x \) is \( \frac{1}{x} \).
  • If you know \( \ln x \), calculating \( \ln \left( \frac{1}{x} \right) \) becomes straightforward using the inverse relationship.
  • For example, if \( \ln 4 = 1.3863 \), then \( \ln \left( \frac{1}{4} \right) = -1.3863 \) because we apply the Reciprocal Rule.
This property simplifies many problems involving logarithms by reducing complex fraction-related logarithmic expressions into straightforward calculations. Essentially, it offers a pathway to solving logarithmic equations where reciprocals are involved.
Logarithm Application
Logarithms are not just abstract mathematical tools; they have a wide range of applications in real life and many fields of study like science and engineering. Understanding their properties, like the reciprocal rule, is critical in unlocking these applications. Here is how logarithmic properties can be applied:
  • In Financial Modeling: Logarithms help in calculating returns and understanding the time value of money. They are used in formulas for compound interest and growth rates.
  • In Biology: Logarithms describe processes such as population decay or growth through models like the logistic growth model.
  • In Physics: They are essential in the calculation of phenomena involving exponential growth or decay, like radioactive decay or the intensity of sound and light over distance.
The ability to use logarithmic properties, including the reciprocal rule, allows for efficient modeling and solving of these real-world problems. In sum, understanding and applying the concepts of logarithms ensures a deeper mathematical insight into natural phenomena and technological developments.

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Most popular questions from this chapter

Differentiate. $$ f(t)=\ln \left|\frac{1-t}{1+t}\right| $$

A beam of light enters a medium such as water or smoky air with initial intensity \(I_{0}\). Its intensity is decreased depending on the thickness (or concentration) of the medium. The intensity I at a depth (or concentration) of \(x\) units is given by $$I=I_{0} e^{-\mu x}$$ The constant \(\mu\left({ }^{u} m u "\right),\) called the coefficient of absorption, varies with the medium. Use this law for Exercises 62 and \(63 .\) Concentrations of particulates in the air due to pollution reduce sunlight. In a smoggy area, \(\mu=0.01\) and \(x\) is the concentration of particulates measured in micrograms per cubic meter \(\left(\mathrm{mcg} / \mathrm{m}^{3}\right)\) What change is more significant-dropping pollution levels from \(100 \mathrm{mcg} / \mathrm{m}^{3}\) to \(90 \mathrm{mcg} / \mathrm{m}^{3}\) or dropping them from \(60 \mathrm{mcg} / \mathrm{m}^{3}\) to \(50 \mathrm{mcg} / \mathrm{m}^{3}\) ? Why?

The profit, in thousands of dollars, from the sale of \(x\) thousand candles can be estimated by $$P(x)=2 x-0.3 x \ln x$$. a) Find the marginal profit, \(P^{\prime}(x)\). b) Find \(P^{\prime}(150),\) and explain what this number represents. c) How many thousands of candles should be sold to maximize profit?

Pharmaceutical firms invest significantly in testing new medications. After a drug is approved by the Federal Drug Administration, it still takes time for physicians to fully accept and start prescribing it. The acceptance by physicians approaches a limiting value of \(100 \%,\) or \(1,\) after \(t\) months. Suppose that the percentage \(P\) of physicians prescribing a new cancer medication after \(t\) months is approximated by $$ P(t)=100\left(1-e^{-0.4 t}\right) $$ a) What percentage of doctors are prescribing the medication after 0 months? 1 month? 2 months? 3 months? 5 months? 12 months? 16 months? b) Find \(P^{\prime}(7)\), and interpret its meaning. c) Sketch a graph of the function.

A student made the following error on a test: \(\frac{d}{d x} e^{x}=x e^{x-1}\) Identify the error and explain how to correct it.
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