Problem 3 - Calculus: An Intuitive and Physical Approach

Question

Look up in the table of natural logarithms the following quantities: (a) \(\log 3 . \quad\) Ans. \(1.0986 .\) (b) \(\log 5 .\) (c) \(\log 10 . \quad\) Ans. \(2.3026\) (d) \(\log 0.1\). (e) \(\log 0.5 .\) Ans. \(-0.6931\).

Step-by-Step Solution

Answer

\(\log 5 \approx 1.6094\). \(\log 0.1 = -\log 10\) hence, \(\log 0.1 = -2.3026\).

Step 1 - Locate \(\log 5\) in the Table

To find \(\log 5\), locate the row in the natural logarithm table that corresponds to the number 5. The value given in the table represents \(\log 5\).

Step 2 - Locate \(\log 10\) in the Table

To verify \(\log 10\), check the natural logarithm table to confirm that the value is \(2.3026\).

Step 3 - Calculate \(\log 0.1\)

Knowing \(\log 10\), use the property \(\log (1/10) = -\log 10\) to calculate \(\log 0.1\). Since \(\log 10 = 2.3026\), \(\log 0.1 = -2.3026\).

Step 4 - Locate \(\log 0.5\) in the Table

To verify \(\log 0.5\), check the natural logarithm table to confirm that the value is \(\log 0.5 = -0.6931\).

Key Concepts

Logarithm Properties

Understanding logarithm properties is pivotal to manipulating and calculating logarithms efficiently. These properties are mathematical rules that govern how logarithms behave.

One fundamental property is that the logarithm of a product equals the sum of the logarithms of the factors: \[\begin{equation}\log(ab) = \log(a) + \log(b)\end{equation}\]Another key property is the ratio rule, where the logarithm of a quotient is the difference between the logarithm of the numerator and the denominator:\[\begin{equation}\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\end{equation}\]For natural logarithms specifically, which use the base 'e', there's also the power rule. This rule says that the logarithm of a number raised to an exponent is the exponent times the logarithm of the base number:\[\begin{equation}\log(a^b) = b\cdot\log(a)\end{equation}\]And importantly, the change of base rule allows the computation of logarithms with different bases using a common base. These properties together make calculations more efficient and are foundational in understanding the nature of logarithmic functions.

Using Logarithm Tables

Before the widespread availability of calculators and computers, logarithm tables were essential tools for mathematicians and scientists. A logarithm table lists the logarithms of many numbers, usually in a base like 10 (common logarithms) or 'e' (natural logarithms).

To use a logarithm table, first, locate the main number (called the mantissa) in a column or row. Then find the corresponding value (called the characteristic) that provides the magnitude of the number. When combined, these pieces give the logarithm value.

For instance, in our exercise, \[\begin{equation}\log(5)\end{equation}\]would be found by looking in the '5' row and matching it with the appropriate column for decimal places. Similarly, inverse operations with logarithms, like finding antilogs (or exponentiation), can also be done using such tables, but it requires a slightly different process.

Natural Logarithm Calculations

The natural logarithm is a logarithm that's based on the number 'e', which is approximately equal to 2.71828. Natural logarithms are denoted as \[\begin{equation}\ln(x)\end{equation}\]or sometimes \[\begin{equation}\log_e(x)\end{equation}\]However, on calculators and in tables, it's often simplified to \[\begin{equation}\log(x)\end{equation}\]when the base 'e' is implied.

In calculations, it is often necessary to manipulate the natural logarithm to solve equations or find values. For example, the natural log of 1 is always 0, and the natural log of 'e' is always 1. These can be written as:\[\begin{equation}\ln(1) = 0\end{equation}\]and\[\begin{equation}\ln(e) = 1\end{equation}\]When a number is not an integer, a table or calculator is necessary for precise values. In our exercise, the natural logarithm can be inverted to find numbers as powers of 'e' through exponentiation, which is part of what makes the natural logarithm so vital in various fields, including calculus, physics, and engineering.