** Logarithmic vs Exponential | Exponential Function vs Logarithmic Function
**

Functions are one of the most important classes of mathematical objects, which are extensively used in almost all subfields of mathematics. As their names suggest both exponential function and logarithmic function are two special functions.

A function is a relation between two sets defined in such a way that for each element in the first set, the value that corresponds to it in the second set, is unique. Let ƒ be a function defined from the set *A* into set *B*. Then for each x *ϵ* *A*, the symbol ƒ(x) denotes the unique value in the set *B* that corresponds to x. It is called the image of x under ƒ. Therefore, a relation ƒ from *A* into *B* is a function, if and only if, for each x*ϵ A* and y *ϵ A*, if x = y then ƒ(x) = ƒ(y). The set *A* is called the domain of the function ƒ, and it is the set in which the function is defined.

**What is exponential function?**

The exponential function is the function given by ƒ(x) = e^{x}, where e = lim( 1 + 1/n) ^{n} (≈ 2.718…) and is a transcendental irrational number. One of the specialties of the function is that the derivative of the function is equal to itself; i.e. when y = e^{x}, dy/dx = e^{x}. Also, the function is an everywhere continuous increasing function having the x-axis as an asymptote. Therefore, the function is one-to-one too. For each x *ϵ R*, we have that e^{x}> 0, and it can be shown that it is onto *R*^{+}. Also, it follows the basic identity e^{x+y} = e^{x}.e^{y} and e^{0 }= 1. The function can also be represented using the series expansion given by 1 + x/1! + x^{2}/2! + x^{3}/3! + … + x^{n}/n! + …

**What is logarithmic function?**

The logarithmic function is the inverse of the exponential function. Since, the exponential function is one-to-one and onto *R*^{+}, a function g can be defined from the set of positive real numbers into the set of real numbers given by g(y) = x, if and only if, y=e^{x}. This function g is called the logarithmic function or most commonly as the natural logarithm. It is denoted by g(x) = log e^{x} = ln x. Since it is the inverse of the exponential function, if we take the reflection of the graph of the exponential function over the line y = x, then we will have the graph of the logarithmic function. Thus, the function is asymptotic to the y-axis.

Logarithmic function follows some basic rules out of which ln xy = ln x + ln y, ln x/y = ln x – ln y and ln xy = y ln x are the most important. This is also an increasing function, and it is continuous everywhere. Therefore, it is also one-to-one. It can be shown that it is onto *R*.

• The exponential function is given by ƒ(x) = e • The domain of the exponential function is a set of real numbers, but the domain of the logarithmic function is a set of positive real numbers. • The range of the exponential function is a set of positive real numbers, but the range of the logarithmic function is a set of real numbers. |

John Ghackie says

Q.E.D. !