In scientific and engineering practice, the natural system of logarithms uses the number 2.718281828459042. Since this number is frequently encountered, the letter e is used. Many natural occurrences can be expressed by exponential equations with e as the base. For example, the decay of radioactive isotopes can be expressed as a natural logarithm equation. These logarithmic expressions are called natural logs because e is the basis for many laws of nature.

The expression In is used to represent a logarithm when e is the base. Therefore, the exponential equation is written as

e^X = N.

and the logarithm expression is written as

As with base 10 logs (common logs), natural logs can be determined easily with the aid of a calculator.

Base 10 logs are often referred to as common logs. Since base 10 is the most widely used number base, the "10" from the designation log_lo is often dropped. Therefore, any time "log" is used without a base specified, one should assume that base 10 is being used.

Anti-Logarithms

An anti-logarithm is the opposite of a logarithm. Thus, finding the anti-logarithm of a number is the same as finding the value for which the given number is the logarithm. If log_lo X = 2, then 2.0 is the power (exponent) to which one must raise the base 10 to obtain X, that is, X = 102.0 = 100. The determination of an anti-log is the reverse process of finding a logarithm.

Example:

Multiply 38.79 and 6896 using logarithms.

Log 38.79 = 1.58872 Log 6896 = 3.83860

Add the logarithms to get 5.42732

Find the anti-log.

Anti-log 5.42732 = 2.675 x 10⁵ = 267,500

Thus, 38.79 x 6896 = 2.675 x 10⁵ = 267,500

Natural and Common Log Operations

The utilization of the log/In can be seen by trying to solve the following equation algebraically. This equation cannot be solved by algebraic methods. The mechanism for solving this equation is as follows:

How would you calculate x in the following equation? log x=5

The easy way to solve this equation is to take the anti-log. As division is the reverse of multiplication, so anti-log is the reverse of log. To take the anti-log log_l0 x = 5:

This is accomplished on a calculator by pressing the 5, INV, then the LOG key. This causes the inverse of the log process.

Summary

The important information in this chapter is summarized below.

Logarithms Summary

A number L is said to be the logarithm of a positive real number N to the base b (where b is real, positive, and not equal to 1), if L is the exponent to which b must be raised to obtain N, or the function can be expressed as

L = Log_b N

for which the inverse is

N = b^L

Simply stated, the logarithm is the inverse of the exponential function.