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The concepts of logarithms and exponentials are crucial for Algebra, and the formulas are also essential for various scientific calculations and current-day applications. Invented by John Napier in 1614, logarithms primarily combine two Greek words, such as ‘logos’ and ‘arithmos.’ It is used in different mathematical calculations to represent the apparent level of variable quantities electromagnetic field strength, sound intensity, and more. If a, x and y are three real numbers, then as per the below equation X=ay,
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we can represent them in logarithmic expression with base a of the logarithm of x such as
loga x = y
a logarithm is primarily the inverse function to exponentiation, and it is widely used by scientists, engineers, and navigators to conduct high-accuracy computations with precision. There are two types of logarithms used, such as common logarithm, which is also known as the base 10 logarithm, and natural logarithm and known as base e logarithm.
On the other hand, an exponential is an expression about the exponent. It involves a variable in an exponent 10x. Logarithms are the opposite of exponentials as division is the opposite of multiplications, or we can also say logarithms are the inverse of exponentials.
We can portray the relationship between exponential and logarithms with the following equation.
Y=bx is equivalent to logb(y) = x
The logarithmic function y = logax is equivalent to the exponential equation x = ay . y = logax where x = ay & a> 0 a≠1
On the other hand, if we consider the inverse of the above equation, then x = ay, and here y equals to logax. So, we can say logarithm is nothing more than an exponent, and we can also simplify an exponent as the number of times it is used in a multiplication. It is written as exp(x) or ex. we can also define a exponential function as y = ex dy/dx = ex
The exponentials functions carry essential weightage in the field of mathematics and calculation. It is also used in all the real-world applications like population growth & falloff, and more.
Logarithms are used to calculate the magnitude of an earthquake on a logarithmic scale; the Richter Scale uses logarithmic formula such as M= log10 A + B adhere A is the amplitude and B is the distance correction which is calculated by the Seismograph. A logarithm is also used to measure sound in decibels, but in a logarithmic scale, it is calculated as dB = 10 log10 (p X 1012. In this p is the sound pressure.
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Logarithms mainly minimize the time required for multiplying numbers with higher digits; it simplifies the log and complex calculations. It makes not only the scientific calculations but also real-life solutions relatively easier.
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