 # 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.

### 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)

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