Logarithmic Functions

In mathematics, scholars used logarithms to change division and multiplication problems into subtraction and addition problems, even before the discovery of calculus. In logarithm to get a certain number, the power is raised to some number which is typically a base number. Logarithmic functions are inverse of exponential functions and you can express any exponential function in logarithmic form. Likewise, all logarithmic function is rewritten in exponential form. When you want to work with very large numbers logarithms are useful in manipulating numbers of a much more manageable size. In this section, we will further discuss the definition, formula, and functions in detail along with some examples.


Logarithmic Functions Definition

A logarithm is an exponent that is written in a special way. For example, we know the following exponential equation is true 3² = 9. Here the exponent is 2 and the base is 3. In the logarithmic form, we will write the equation as log3 9 = 2. In words we say this as “the logarithm of 9 to the base 3 is 2”. Here we have effectively moved the exponent down to the mainline. This was done to make divisions and multiplications easier but still, logarithms are very handy in mathematics.

A logarithmic function is defined as f (x) = logb x. here the base of logarithm is b. base e and base 10 are the common bases we use which we meet in logs to base 10 and natural logs.

In real life, there are many logarithmic applications like in electronics, earthquake analysis, acoustic, and population prediction. 


Introduction to Common and Natural Logarithmic Function

Common Logarithmic Function

A logarithm with base 10 is a common logarithm. In our number system, there are ten bases and ten digits from 0-9, here the place value is determined by groups of ten. You can remember common logarithms with the one whose base is common as 10. 

Natural Logarithmic Function

A natural logarithm is different. When the base of the common logarithm is 10 the base of a natural logarithm is number e. Although e represents a variable it is a fixed irrational number that equals to 2.718281828459. Sometimes e is also known as Euler’s number or Napier’s constant. The letter e is chosen to honour mathematician Leonhard Euler. e looks complicated but is rather an interesting number. The function f (x) = loge x has multiple applications in business, economics, and biology. Therefore, e is an important number.


Properties of Logarithmic functions

The Product Rule

Logb (MN) = logb (M) + logb (N)

This property denotes that logarithm of a product is the sum of the logs of its factors.

Multiply two numbers having same base, then add the exponents

Example: log 20 + log 2 = log 40

The Quotient Rule

Log(M/N) = logb  (M) – log(N)

This property denotes that the log of a quotient is the difference of the log of the dividend and the divisor.

Divide two numbers having the same base and subtract the exponent.

Example: log6 54 – log9 = log6 (54/9) = log6 6 = 1

The Power Rule

Log(Mp ) = p log(M)

The property denotes that log of a power is the exponent times the logarithm of the base of the power.

Raise an exponential expression to power and multiply the exponents

4 log5 (2) 

= log5 (24

= log5 (16)

The Zero Exponent Rule

Log1 = 0

Change of Base Rule

log(x) = x / b or log(x) = log10 x / log10 b

some other properties of logarithms are as follows:

logb (xy) = logx + logy

logb (x/y) = logb x – logy

logb (xr ) = rlogb x

if logb x = logb y therefore x = y


Examples of Logarithmic Functions


1. Write the exponential equation in logarithmic form

  1. 52 = 25

  2. 4-3  = 1/64

  3. (1/2)-4   = 16

Solution: a. 52  = 25  becomes 2 = log25

b. 4-3  = 1/64 becomes -3 = log4  (1/64)

c. (1/2)-4  = 16 becomes -4 = log(1/2)  16


2. Write logarithmic equation in exponential form

a. log6 36 = 2

b. loga m = p

solution: a. log6 36 = 2 become 62 = 36

b. loga m = p becomes a= m


3. Solve the following equations

a. log7 49 = y

b. log(1/8) = y

solution: a. log7 49 = y becomes 7y  = 49

since, 49 = 72         7= 7

so, y = 2

b. log2 (1/8) = y becomes 2y = 1/8

since, 1/8 = 2-3    2y = 2-3 

so, y = -3

FAQ (Frequently Asked Questions)

1. What is the parent function of logarithmic functions?

Logarithms are used to explore properties of exponential functions and to solve exponential functions. For any log parent function is written as f(x) = logb x. for example, g(x) = logx corresponds to another family of functions then h(x) = log8 x. the example graphs the common log: f(x) = log x. you can change the log into exponential. Finding the parent of logarithmic functions is valuable in calculus where it is used to calculate the slope of certain functions and the area bounded by certain curves.

2. What does a logarithmic function look like?

While speaking a logarithmic function is said as “the log, base a, of x.” the logarithmic function is the inverse of exponential function, so logarithms can only be expressed by using its exponential form. It is the same operation by thinking “a to the power y equals to x”. The common logarithmic function is written as y = log x and natural logarithmic function is written as f (x) = log e x. Here ‘e’ is any number whose value is greater than zero, but excludes ‘1’. An interesting fact is that the logarithmic function is always on the positive side of the y-axis. It never crosses over to reach the negative side of the y-axis.