# Logarithm

### What is logarithms

The use of logarithms in industry is so extensive that a clear understanding of their purpose and use is an absolute necessity. At first exposure, logarithms often appear vague and mysterious due to the mathematical operations required to find the logarithm and antilogarithm using the longhand table approach that is typically taught in mathematics courses. However, almost all of today’s scientific calculators have the common and natural log functions, eliminating the complexity of applying logarithms and allowing us to concentrate on the positive characteristics of the function.

### Basic Relationships

Let us first examine the relationship between the variables of the logarithmic function. The mathematical expression
$$N=(b)^{x}$$
states that the number $N$ is equal to the base $b$ taken to the power $x$. A few examples:
\begin{aligned}100 &=(10)^{2} \\27 &=(3)^{3} \\54.6 &=(e)^{4} \quad \text { where } e=2.7183\end{aligned}
If the question were to find the power $x$ to satisfy the equation
$$1200=(10)^{x}$$
the value of $x$ could be determined using logarithms in the following manner:
$$x=\log _{10} 1200=\mathbf{3 . 0 7 9}$$
revealing that
$$10^{3.079}=1200$$
Note that the logarithm was taken to the base 10, the number to be taken to the power of $x$. There is no limitation to the numerical value of the base except that tables and calculators are designed to handle either a base of 10 (common logarithm, LOG) or base $e=2.7183$ (natural logarithm, IN ). In review, therefore, If $N=(b)^{x}$, then $x=\log _{b} N$ The base to be employed is a function of the area of application. If a conversion from one base to the other is required, the following equation can be applied:
$$\log _{e} x=2.3 \log _{10} x$$
The content of this chapter is such that we will concentrate solely on the common logarithm. However, a number of the conclusions are also applicable to natural logarithms.

### Some Areas of Application

The following is a short list of the most common applications of the logarithmic function:
• This chapter will demonstrate that the use of logarithms permits plotting the response of a system for a range of values that may otherwise be impossible or unwieldy with a linear scale.
• Levels of power, voltage, and the like, can be compared without dealing with very large or very small numbers that often cloud the true impact of the difference in magnitudes.
• A number of systems respond to outside stimuli in a nonlinear logarithmic manner. The result is a mathematical model that permits a direct calculation of the response of the system to a particular input signal.
• The response of a cascaded or compound system can be rapidly determined using logarithms if the gain of each stage is known on a logarithmic basis. This characteristic will be demonstrated in an example to follow.
There are a few characteristics of logarithms that should be emphasized:
1. The common or natural logarithm of the number 1 is 0.
$\log _{10} 1=0$ just as $10^{x}=1$ requires that $x=0$.
2. The log of any number less than 1 is a negative number:
$$\begin{array}{l}\log _{10} 1 / 2=\log _{10} 0.5=-0.3 \\\log _{10} 1 / 10=\log _{10} 0.1=-1\end{array}$$
3. The log of the product of two numbers is the sum of the logs of the numbers.
$$\log _{10} a b=\log _{10} a+\log _{10} b$$
4. The log of the quotient of two numbers is the log of the numerator minus the log of the denominator.
$$\log _{10} \frac{a}{b}=\log _{10} a-\log _{10} b$$
5. The log of a number taken to a power is equal to the product of the power and the log of the number.
$$\log _{10} a^{n}=n \log _{10} a$$
Example 1: Evaluate each of the following logarithmic expressions:
a. $\log _{10} 0.004$
b. $\log _{10} 250,000$
c. $\log _{10}(0.08)(240)$
d. $\log _{10} \frac{1 \times 10^{4}}{1 \times 10^{-4}}$
e. $\log _{10}(10)^{4}$
Solution:
a. $-2.398$
b. $+5.398$
c.
$$\log _{10}(0.08)(240)=\log _{10} 0.08+\log _{10} 240=-1.097+2.380 =1.283$$
d.
\begin{aligned} \log _{10} \frac{1 \times 10^{4}}{1 \times 10^{-4}} &=\log _{10} 1 \times 10^{4}-\log _{10} 1 \times 10^{-4}=4-(-4) \\ &=8 \end{aligned}
e.
$$\log _{10} 10^{4}=4 \log _{10} 10=4(1)=4$$