Logarithmic Functions 

In mathematics, the logarithmic function is the inverse of the exponential function. It is also a multiplicative function. We can express the logarithm of the absolute value of any number as the power to which another fixed value, the characteristic, must be raised to produce that number.

For example, the logarithmic function of y = logax is equal to x = ay. Here, y = logax is the logarithmic form. However, for it to be equal, there are some terms and conditions:

• x = ay
• a > 0
• a≠1

POINTS TO REMEMBER:

• loga1 = 0
• log10 = 1
• log10 = 1
• loga(ax) = x
• alogax = x
• log(bc) = logb + logc
• log(b/c) = logb – logc
• log(xd) = d logx

If you are unfamiliar with common logarithms, here is a quick refresher: 

• With base 10, numbers to the right of the decimal point indicate powers of ten.
• With base e, numbers to the right of the decimal point indicate exponents.

Common Logarithmic Functions

log 10 N or log N denote the common logarithmic function, as these logarithmic functions make it easier for the student to understand it. 

Common Logarithmic Functions Properties

There are some basic properties of common logarithms, which are as follows :

1. Product Rule: 

If ​a​ and ​b​ are two common logarithmic functions, then the product of ​a​ and ​b​ equals the sum of the products of ​a​ and ​b​.

log (ab) = log a + log b

1. Quotient rule: 

In common logarithms, division by the same number is equivalent to subtraction of that number from both values.

log(m/n) = log m – log n

1. Power rule:  

Assume a number is raised to a power. The log of the number is equal to the product of the exponent and its natural log.

log (mn) = n log m

Natural Logarithmic Functions

In mathematics, the natural logarithm is the logarithm to the base e. In other words, it is the exponent to which we have to raise e to equal number N. Multiple logarithms are also known as logarithmic functions, and we usually write with an ‘ln’ prefix. 

Calculators have keys that represent both common and natural logarithms. The key for the natural log is labeled “e” or “ln”, while that of the common logarithm is labeled “log”.

Properties of Natural Logarithmic Functions

There are some basic properties of natural logarithms, which are as follows :

1. Product Rule: 

If ​a​ and ​b​ are two natural logarithmic functions, then the product of ​a​ and ​b​ equals the sum of the products of ​a​ and ​b​.

ln (ab) = ln a + ln b

1. Quotient Rule: 

In natural logarithms, division by the same number is equivalent to subtraction of that number from both values.

ln (a/b) = ln (a) – ln (b)

1. Reciprocal Rule: 

In the reciprocal rule in the natural logarithmic function, the log function will be inverted.  

ln (1/a) = −ln (a)

Logarithmic Functions Properties

The logarithm and exponential functions are opposites of each other. As a result, we can easily deduce that if a certain number is to the power of b, then that same number represented in logarithmic form is equal to b. 

Now, we will discuss the logarithmic function properties and how they are helpful.

1. The domain of the logarithmic functions is all real numbers (0, +∞).
2. The range of the logarithmic function is from negative infinity to positive infinity (-∞,+∞).
3. The logarithmic function has wide applications in mathematics. It possesses some measure of continuity and differentiability.
4. The points with the coordinates (1,0) and (b,1) will always be on the graph of the function logb of x.

Exponential Functions

An exponential function is merely a function in which the variable occurs as an exponent. 

Suppose we have a function y=f(x) where x is the independent variable and y is the dependent variable. Then, whenever x occurs as an exponent to some power  ‘a’ (denoting the base) in y, the whole expression is known as an exponential function.

f(x)= ax

Here, 

• x is a variable and is a real number. But if the variable is negative, then the function won’t be defined for -1 < x < 1
• a is known as constant and is the base of the function. (a>0 but should not be equal to 1) 

Properties of Natural Exponential Functions

There are some basic properties of natural exponents, which are as follows:

1. Product rule: 

If two exponential functions are in multiplications, then their powers will add up. 

1. xa. xb = x(a+b) 

1. Power rule: 

If two powers are present on a function, we multiply them. 

(am)n = am*n

1. Quotient rule: 

If two bases have the same power in a division equation, we subtract the powers. 

am/an = a(m-n) 

Properties of Exponential Functions

There are some basic properties of exponential functions, which can help students understand the questions better. These are :

1. The domain of the exponential function is (-∞,+∞), and it is defined for all real numbers x.
2. The exponential function is also called exponential growth. It is continuous and differentiable, and its range is (0, + ∞).
3. The coordinates of the points (0,1) and (1, a) are ALWAYS on the line y =  ax.
4. The fact that e is equal to 2.7182818… can be proven by dissecting the function into pieces of the exponential function.

Conclusion

This article provides an insight into logarithmic and exponential functions. It gives the formulas related to logarithmic and exponential functions that help understand the differences between both methods. Moreover, the knowledge of exponential and logarithmic functions will help the students solve the problem related to the topic easily