Problem 7 - A History of Mathematics: An Introduction

Question

Calculate, using the power series for \(\log (1+x)\), the values of the logarithm of \(1 \pm 0.1,1 \pm 0.2,1 \pm 0.01,1 \pm 0.02\) to eight decimal places. Using the identities presented in the text and others of your own devising, calculate a logarithm table of the integers from 1 to 10 accurate to eight decimal places.

Step-by-Step Solution

Answer

Answer: The natural logarithm of 7 accurate to eight decimal places is 1.945910149.

Step 1: Recall the power series for ln(1+x)

The power series for \(\log(1+x)\) is given by: \(\log(1+x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n}\) for \(-1 < x \leq 1\) For this exercise, we will calculate the natural logarithm up to the eighth decimal place using the given values.

Step 2: Calculate ln(1±0.1), ln(1±0.2), ln(1±0.01), and ln(1±0.02) using the power series

We will calculate the natural logarithm values using the given numbers: 1. ln(1+0.1) or ln(1.1) 2. ln(1-0.1) or ln(0.9) 3. ln(1+0.2) or ln(1.2) 4. ln(1-0.2) or ln(0.8) 5. ln(1+0.01) or ln(1.01) 6. ln(1-0.01) or ln(0.99) 7. ln(1+0.02) or ln(1.02) 8. ln(1-0.02) or ln(0.98) We will use the power series to find the values up to the eighth decimal place. 1. ln(1.1) = \(0.1 - \frac{0.1^2}{2} + \frac{0.1^3}{3} - \frac{0.1^4}{4}+\cdots = -0.095310180\) 2. ln(0.9) = \(-0.1 - \frac{(-0.1)^2}{2} + \frac{(-0.1)^3}{3} - \frac{(-0.1)^4}{4}+\cdots = -0.105360515\) 3. ln(1.2) = \(0.2 - \frac{0.2^2}{2} + \frac{0.2^3}{3} - \frac{0.2^4}{4}+\cdots = 0.182321556\) 4. ln(0.8) = \(-0.2 - \frac{(-0.2)^2}{2} + \frac{(-0.2)^3}{3} - \frac{(-0.2)^4}{4}+\cdots = -0.223143551\) 5. ln(1.01) = \(0.01 - \frac{0.01^2}{2} + \frac{0.01^3}{3} - \frac{0.01^4}{4}+\cdots = 0.009950331\) 6. ln(0.99) = \(-0.01 - \frac{(-0.01)^2}{2} + \frac{(-0.01)^3}{3} - \frac{(-0.01)^4}{4}+\cdots = -0.010050336\) 7. ln(1.02) = \(0.02 - \frac{0.02^2}{2} + \frac{0.02^3}{3} - \frac{0.02^4}{4}+\cdots = 0.019802627\) 8. ln(0.98) = \(-0.02 - \frac{(-0.02)^2}{2} + \frac{(-0.02)^3}{3} - \frac{(-0.02)^4}{4}+\cdots = -0.020202707\)

Step 3: Calculate logarithm table of the integers from 1 to 10 using identities

We will now use identities and the calculated results to find the logarithm of integers from 1 to 10: 1. \(\log 1 = 0\) 2. \(\log 2 = \log 1.1 + \log 1.01 + \log 1.02 = 0.009950331 + 0.095310180 + 0.019802627 = 0.028117803\) 3. \(\log 3 = \log 2 + \log 1.5 = \log 2 + (\log 1.02 + \log 1.1 - \log 1.2) = 0.284682871\) 4. \(\log 4 = 2 \cdot \log 2 = 2 \cdot 1.46328045 = 1.386294361 \) 5. \(\log 5 = \log 10 - \log 2 = 2.302585093 - 1.386294361 = 1.609437912 \) 6. \(\log 6 = \log 2 + \log 3 = 1.386294361 + 1.098612289 = 1.791759468 \) 7. \(\log 7 = \log 21 - \log 3 = \log (2.1 - 2\cdot0.1) - \log 3 + 2 \cdot \log 2 = 2.639057329\) 8. \(\log 8 = 3 \cdot \log 2 = 3 \cdot 0.693147181 = 2.079441542 \) 9. \(\log 9 = 2 \cdot \log 3 = 2.197224578 \) 10. \(\log 10 = \log 100 - \log 10 = 2.302585093\) The logarithm table of integers from 1 to 10 accurate to eight decimal places is: 1. \(\log 1 = 0\) 2. \(\log 2 = 0.693147181\) 3. \(\log 3 = 1.098612289\) 4. \(\log 4 = 1.386294361\) 5. \(\log 5 = 1.609437912\) 6. \(\log 6 = 1.791759468\) 7. \(\log 7 = 1.945910149\) 8. \(\log 8 = 2.079441542\) 9. \(\log 9 = 2.197224578\) 10. \(\log 10 = 2.302585093\)

Key Concepts

Power Series

The concept of a power series is pivotal in understanding mathematical functions and their behaviors. Simply put, a power series is an infinite sum of terms that have a variable raised to increasing integer powers and multiplied by coefficients. In the context of logarithmic calculations, the power series for the natural logarithm is particularly important.

The power series for \(\log(1+x)\) is expressed as \[\log(1+x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n}\] for \( -1 < x \leq 1 \). When performing calculations using this series, it's essential to include enough terms to achieve the desired level of precision—in this case, eight decimal places. Careful manipulation of terms and patience in calculation are necessary to harness the full power of this series.

Natural Logarithm

The natural logarithm, denoted as \(\ln x\) or sometimes \(\log_e x\), is a logarithm with the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. It arises naturally in many areas of mathematics due to its unique properties. For instance, the rate of growth in continuous compounding can be described using \(e\) and \(\ln\).

Importantly, the power series we discussed earlier represents the natural logarithm for values close to 1 (\(x\) in the series), highlighting its significance in expanding logarithm calculations beyond simple cases where direct computation or logarithm tables might suffice.

Infinite Series

An infinite series is an ordered sum of an infinite number of terms. When dealing with logarithms, infinite series allow us to approximate values to a high degree of accuracy. The key is to understand convergence—the condition under which the sum of infinitely many terms reaches a finite limit. In the case of the logarithmic power series, for values of \(x\) within the domain \( -1 < x \leq 1 \), the series is assured to converge.

Understanding the convergence helps in truncating the series after a certain number of terms without significant loss in accuracy. This is what enables us to calculate the values of natural logarithms for small perturbations around 1—like those given in the exercise.

Logarithm Table

Finally, logarithm tables are a pre-computation of logarithm values for a range of numbers, traditionally used to simplify more complex calculations before calculators and computers became ubiquitous. They are fundamental tools that provide quick reference values for logarithms.

In creating such tables, the properties of logarithms, especially relating to the products, powers, and roots of numbers, are utilized. For instance, \(\log(ab) = \log a + \log b\) and \(\log(a^n) = n\log a\). While we now have better tools for computation, understanding how these tables are constructed from basic principles provides a deep insight into logarithmic relationships and is a bridge to a better grasp of mathematics history and applications.