Logarithms-Common Logarithms Characteristics

In terms of mathematics, the logarithm is said to be an inverse function to exponentiation. A logarithm is used to represent larger quantities in smaller forms. This article revolves around the basic concept of the logarithm, common logarithm characteristics, functions and rules of the logarithm. Some questions based on this topic are also discussed in this article.

Definition of logarithm

The logarithm can be defined as the exponential power to which a number has to be raised to get a realistic mathematical value. The logarithm is convenient to express larger numbers into smaller forms. A logarithm is used to prove logarithm rules of multiplication and division in form of a logarithm of addition and subtraction. In logarithmic representation, b must be raised to some power that gives “a” as value. In the mathematical formula, it can be represented as:

by = a 

Or, logba = y

Here, a and b must be two positive real numbers, 

y = real number,

In terms of the logarithm, “a” is called an argument, which must lie inside the log, and “b” is called the base, which lies at bottom of the log.

Types of Logarithm

Logarithms are primarily of two types:

• Common Logarithms: The base-10 logarithm is a generic term for common logarithms characteristics. The mathematical notation is log10. For instance, log1055
• Natural Logarithms: The base-e logarithm is the name given to natural logarithms. The symbol for this is loge. For instance, loge250

Functions of Logarithms

The fact that logarithms are beneficial in solving exponential equations accounts for a large part of their strength. Sound, earthquakes, the glory of the stars, and chemistry are just a few instances of what I’m talking about. To understand the real-life situation of logarithms, one can begin by measuring the strength of an earthquake. To do so, first and foremost, one must be familiar with earthquake measurements, which are made possible by a device known as a seismograph that records the output of any seismic activity that occurs.

Common Characteristics of Logarithms

Unlike exponential derivative functions, these logarithmic functions are reverses of such exponentials that are considered as common logarithms characteristics. For instance, the inverse of the exponential function h = cg is g = ah. The logarithmic operation k = logij is determined to be identical to this equation of exponential as j = ik.

The logarithm of a digit larger than 1 has a positive characteristic and has fewer digits than the number’s integral. The logarithm of a real integer less than 1 has a negative characteristic and is mathematically higher by 1 than that of the multitude of zeroes between both the numeric sign and its first significant figure.

Rules of Logarithm

It is possible to derive the basic rules for using logarithm function logqp from basic rules for using exponents’ qx because taking an inverse logarithm is the reverse of exponentiation. For simplicity, the rules are written below in representations of the natural type logarithm ln(p). The constraints used for any logarithm logqp. Besides that, one has to substitute any occurrence of e with the new base q.

The various rules of logarithms are discussed below:

• Product Rule: It’s possible to derive the product rule for logarithms by using a rule of product for exponentiation. Its formula is: ln(pq) = ln(p) + ln(q)

• • Rule of Power Function: Log of power can be obtained by following the rule for a given power: ln(pq) = qln(p)

• Quotient Rule: This rule of a logarithm is followed by this formula: ln(p/q) = ln(p) − ln(q)

• Log of One: This logarithm of one’s formula is derived from the power of zero’s formula, which is: ln(1) = 0
• Rule of Reciprocal: The reciprocal logarithm rules are derived from the power of the negative one rule, which is: ln(1/p) = −ln(p)
• Exponential Rule: The log of e’s formula is derived from the power of one’s formula, which can be denoted as ln(e) = 1

Conclusion

Most of what humans know today about the logarithm as a feature is quite different from what it was initially intended to be. However, the logarithm rules had become far more than a productive way to calculate with large, unwieldy numbers through the task, consideration, and advancement of many mathematicians. It developed into a stand-alone mathematical relationship and function. The logarithm has grown from a time-saving method to one of essential characteristics in mathematics with time. Modern mathematics relies heavily on this concept, which has been developed to possess complex and negative numbers. It plays a crucial role in theories and is crucial to trigonometry because of its simple processes and visual appeal in various integrals.