Logarithmic and Exponential Functions

Logarithmic and exponential functions are quite significant aspects of mathematics. This stands true both in mathematical theory and practice. It is possible to express any exponential function in the logarithmic form. Similarly, you can write down all logarithmic functions in the exponential form. An exponential equation is one in which the appearance of the variable takes place in an exponent. In contrast, in logarithmic equations, the logarithm of an expression involves a variable. Keep on reading these study material notes on logarithmic and exponential functions to build a formidable understanding of this topic.

Logarithmic functions

The first thing to focus on in this logarithmic and exponential functions study material is the logarithmic functions. Logarithmic functions, simply speaking, happen to be the inverse of exponential functions. This is the direct relationship of logarithmic and exponential functions. In logarithmic functions, to get a number, you must raise the power to some number, known as the base number.
Logarithms are very helpful in letting us operate with numbers that are very large. Moreover, with logarithms, we can manipulate numbers that are of a significantly manageable size.
The inverse of the exponential function y = a^x is x = a^y. Logarithmic function is defined as y = log_ax and it is equivalent to the exponential function x = ay.
The function y = log_ax is only due to the following:

• x = a^y
• a > 0
• a≠1

The base of such a logarithmic function is a.
Always remember that the logarithm is, in its very essence, an exponent. When defining, a^log_a^x = x stands true for every real x > 0.
Below are some useful properties of logarithms presented by these study material notes on logarithmic and exponential functions in accordance with a > 0 and x > 0.

• log_a1 = 0
• log_aa = 1
• log_a(a^x) = x
• a^log_a^x = x
• log_a(bc) = log_ab + log_ac
• log_a() = log_ab – log_ac
• log_a(x^d) = d log_ax

The logarithm of a number is the exponent in its simplest form. This indicates the logarithmic and exponential functions relationship.

Exponential functions

An exponential function refers to a function in which the independent variable happens to be an exponent. Generally speaking, such functions take the following form:
y = f (x) = a^x
Here:

• a > 0
• a≠1,
• x refers to real number

The base of the function is a.
Below are some useful properties of exponents.

• a^-x = 1/a^x
• a^x+y = a^x×a^y
• a^x-y = a^x/a^y
• a⁰ = 1
• a^x = a^y

The above would be true only if x = y

Relationship between logarithmic and exponential functions

No logarithmic and exponential functions study material can be complete without discussing the relationship between the two. An exponential function is given below:
f(x) = b^x
Where,
b > 0
b ≠ 1
A logarithmic function is in the following form:
g(x) = log_b(x)
Where,
b > 0
b ≠ 1
This shows the existence of a relationship between logarithmic and exponential functions.
As you can see, both have the same restrictions on b. This is due to the fact that they both are inversely related. What this implies is that for the same value of b,
b^logb = x
Where,
x > 0
log_bb^x = x
The logarithmic function f(x) = log(x) is a special way of representing the following:
f(x) = log¹⁰(x)
Furthermore, g(x) = lnxg(x) = ln x is a special way of representing the following:
g(x) = log_e(x)
Also, log(x) is referred to as the common logarithm, while the natural logarithm is ln(x).
Also, e is a number wherein
e ≈ 2.71828e
As such, e is an irrational number and so you cannot write it as a fraction of a whole number.

Conclusion

Logarithmic and exponential functions are significant aspects of mathematics. Logarithmic functions are the inverses of functions that are of an exponential nature. In an exponential equation, the appearance of the variable takes place in an exponent. In contrast, in a logarithmic equation, the expression’s logarithm involves a variable. It is important to study the logarithmic and exponential functions along with their properties. Most importantly, you must understand the relationship between these two types of functions.