Logarithmic Functions

Introduction

A logarithmic function is a crucial tool in mathematical calculations. John Napier, a Scottish mathematician, scientist, and astronomer, discovered logarithms in the 16th century. It does have a wide range of uses in astronomical and scientific calculations that involve large numbers. Exponential functions are closely connected to logarithmic functions, and the inverse of the exponential function is frequently used. The exponential function ax = N is transformed to a logarithmic function loga N = x. 

When expressed in exponential form, the logarithm of any number X is the exponent to which the base of the logarithm must be raised to achieve the number X. 

What are Logarithmic Functions and How do they Work?

An exponential function is used to create the logarithmic function. Logarithmic functions can be used to conveniently determine some of the non-integral exponent values. Finding the value of x in the exponential expressions  2x= 8, 2x = 16 is easy, but finding the value of x in 2x = 10 is difficult. 2x = 10 can be transformed into logarithmic form as log2 10 = x using logarithmic functions after that find the value of x. The logarithm counts the number of times the base appears in successive multiples. The following is the formula for converting an exponential function to a logarithmic function.

Exponential function : ax = N

Logarithmic function : loga N = x

The form’s of exponential function could be converted to a logarithmic function. The logarithms were usually calculated using a base of ten, and a Napier logarithm table could be used to find the logarithmic value of any number. For positive whole integers, fractions, and decimals, logarithms could be calculated, but not for negative values.

Natural Logarithms and Common Logarithms

Depending on the base of the logarithms, the logarithms are divided into two types. Natural logarithms but also common logarithms are two types of logarithms. Logarithms to the base ‘e’ are natural logarithms, whereas logarithms to the base of 10 were common logarithms. Further logarithms could be calculated with any base, however, they are most commonly calculated with bases of ‘e’ or ’10’. Natural logarithms are written as loge x and the common logarithms are written as log10 x. The power to which e must be raised to acquire x is equivalent to the value of x obtained using natural logarithms. 

ie e1. 6.9 = 5.

e = 2.718

loge N = 2.303 × log10 N

loge N = 0.4343 × loge N

The value of e is 2.718281828459, however, it is commonly abbreviated as e = 2.718. The formulas above also help in the interconversion of natural and common logarithms.

Properties of Logarithmic Functions

When working with complex logarithmic functions, logarithmic function features prove useful. All common arithmetic operations on numbers are turned into a new set of operations within logarithms. Under logarithmic functions, the product of two numbers equals the sum of the logarithmic values of the two functions. Likewise, division operations were translated into the difference of the two numbers’ logarithms. In the following sections, the properties of logarithmic functions are discussed.

• log(ab) = loga + logb
• log(a/b) = loga – logb
• logax = x loga
• log1 a = 0
• loga a = 1

Derivative of Logarithmic Functions

The slope of the tangent to the curve depicting the logarithmic function is determined by the logarithmic function’s derivation. A simple variable or a trigonometric ratio could be used as a function within the logarithm. The inverse of the logarithmic function is obtained by taking the derivative of the logarithmic function. The derivative of a logarithmic function is calculated using the following formula.

d/dx. Logx = 1/x

Relation Between Exponential and Logarithmic Functions

The logarithmic and exponential functions are inverses of each other, as we’ve already shown. You may now confirm this by looking at the properties. 

• The two functions’ range but also domains have been exchanged

• The points (0,1) and (1, a) are always on the graph of the exponential function, whereas (1,0) and (b,1) are always on the graph of the logarithmic function

• The exponential and logarithmic functions’ product and quotient rules are inextricably linked

Conclusion

The inverses of exponential functions are logarithmic functions, and also any exponential function could be represented in logarithmic form. All logarithmic functions could be rewritten in the exponential form in the same way. Logarithms are extremely important because they allow us to work with very large quantities while managing numbers that are much smaller. It evolved into its own mathematical relation and purpose. The logarithm has progressed from a labour-saving device to one of mathematics’ essential functions. It is now used in many modern areas of mathematics and it has been expanded to include negative even complex numbers.