Problem 40 - Calculus and its applications

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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 \(\ln \frac{1}{4}\):\[\ln \frac{1}{4} = -\ln 4 = -1.3863\]

Key Concepts

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.