Problem 4 - Calculus: An Intuitive and Physical Approach

Question

Find the numbers whose natural logarithms are the following: (a) \(-0.5108\). (b) \(0.7885\). Ans. \(2.2\) (c) \(3.4012\). (d) \(4.3820\). Ans. 80 .

Step-by-Step Solution

Answer

The numbers whose natural logarithms are given are: (a) approximately 0.60. (b) 2.2 (given). (c) approximately 30.01. (d) 80 (given).

Step 1 - Understanding the Concept of Natural Logarithms

The natural logarithm, denoted as \(ln(x)\), is the logarithm to the base \(e\), where \(e\) is the Euler's number approximately equal to 2.71828. To find the number whose natural logarithm is given, we need to find the exponent to which \(e\) must be raised to obtain that number. This can be represented as \(e^{ln(x)} = x\).

Step 2 - Calculate the Number for (a) \(-0.5108\)

To find the number whose natural logarithm is \(-0.5108\), we exponentiate \(e\) to the power of \(-0.5108\): \[x = e^{-0.5108}\]. Calculating this using a scientific calculator or computer software gives us the number.

Step 3 - Calculate the Number for (b) \(0.7885\)

For part (b), the answer has been provided as \(2.2\). We can verify this by checking if \(e^{0.7885} = 2.2\) using a scientific calculator or relevant software.

Step 4 - Calculate the Number for (c) \(3.4012\)

To find the number whose natural logarithm is \(3.4012\), we exponentiate \(e\) to the power of \(3.4012\): \[x = e^{3.4012}\]. Like before, we calculate this number using a calculator or software.

Step 5 - Calculate the Number for (d) \(4.3820\)

For part (d), the answer has been provided as 80. To verify this, we exponentiate \(e\) to the power of \(4.3820\) and see if the result matches: \[x = e^{4.3820} = 80\].

Key Concepts

Euler's number

Euler's number, often denoted as 'e', is one of the most important constants in mathematics. It's approximately equal to 2.71828 and serves as the base for natural logarithms. Why is it so crucial? Well, 'e' has unique properties that make it the go-to constant in the realms of calculus and complex analysis. It's the rate of growth shared by all continually growing processes. For example, if you have one dollar in a bank account that offers 100% interest rate compounded continuously, after one year, you'd have 'e' dollars.

When dealing with natural logarithms, understanding 'e' is non-negotiable. The natural logarithm is simply the power to which 'e' must be raised to obtain a certain number. In our exercise, for instance, to find the number with a natural logarithm of -0.5108, we raise 'e' to that power. The magic of 'e' makes this process knit tightly with exponential growth and decay models found in science and finance.

Exponentiation

Exponentiation is a mathematical operation, involving two numbers, the base 'b' and the exponent 'n'. In essence, it's repeating multiplication of the base, 'b', 'n' times. When 'n' is positive, it's straightforward - for example, if we have 2 raised to the power of 3, written as \(2^3\), it simply means 2 multiplied by itself 3 times, equals 8. Negative exponents, though, flip things a bit; they represent division. So, \(2^{-3}\) is the same as dividing 1 by 2 three times.

In our log exercise, we need to grasp how exponentiation relates to logarithms. Natural logarithms, remember, are all about finding what power 'e' has been raised to. If we know the natural log of a number, we can find the actual number by raising 'e' to the power of that natural log. It's like reverse-engineering the operation. Hence, for the given natural logarithm of 0.7885, exponentiating 'e' by this number gives us the original number, which is (b) 2.2.

Scientific Calculator

A scientific calculator is a student's best friend when dealing with complex mathematical operations like exponentiation, especially when the base is an irrational number like Euler's number 'e'. It's essential for quickly solving problems involving exponents, logarithms, and trigonometry among others.

When using a scientific calculator to solve problems like our textbook exercise, you enter the given natural logarithm value, and then use the calculator's exponentiation function to raise 'e' to that logarithm. Voilà! The calculator provides the number you're looking for. In the context of our exercise, this tool helps us affirm the answers provided, such as \(e^{4.3820} = 80\), ensuring accuracy and understanding in our solve.