As you know, multiplication is a shortcut for addition.

For example,

5 * 3 = 5 + 5 + 5.

Exponents are a shortcut for multiplication. For example,

53 = 5* 5* 5.

Likewise, a logarithm is a shortcut for exponents.

In this section you will learn some simple laws of logarithms. Logarithms are very useful in such calculations. They make even difficult calculations quite easy.

Defining logarithms

We learned that 4²=16 . We can also write this another way :

Log⁴16 = 2

This is a log with subscript of 4. The equation is read as “the log to the base 4 of 16 is 2”.

The log to the base x of y is the number you raise x to, to get y. Thus, the logarithm of a number to a given base is the index to which the base should be raised to get the given number.

Definition

If N and a, a ≠1 are any two positive real numbers and for some real x, if ax = N then x is said to be the logarithm of N to the base “a’, and is written as log_aN = x.

Remember that logarithms are defined only for positive real numbers.

Also, there exists a unique x that satisfies the equation ax = N.

So, log_aN is also unique.

       Exponential function       Logarithmic function

                a^x = N                         x = log_aN

                b^y = N                         y = log_bN

                x^z = Z                         y = log_xZ

Functions defined by such equations are called logarithmic functions.

We can express exponential forms in logarithmic form.

     Exponential form              Logarithmic form

             2⁴ = 16                         4 = log₂16

            1/9 = 1/3²                    –2 = log₃1/9

                 = 3^-2              

If a^x = N1 (a ≠1, a > 0),

then x = log_aN

Observe the following examples:

2⁶= 64 can be written as log₂64 = 6

4³ = 64 can be written as log₄64 = 3

From these examples, we know that logarithms of the same number, i.e., 64, with two different bases, i.e., 2 and 4, are different.

Therefore, the logarithms of the same number to different bases are different.

Try these questions

I     Write the following in logarithmic forms.

1. 2⁴ = 16
2. 10⁴ = 10000
3. 3³ = 27
4. 10^-1 = 0.01
5. 6³ = 216

Answers to questions

1. 2⁴ = 16
   log₂16 = 4


2. 10⁴= 10000
   log₁₀10000 = 4


3. 3³ = 27
   log₃27 = 3


4. 10^-1 = 0.01
   log₁₀0.01 = – 1


5. 6³ = 216
   log₆216 = 3



II     Express each of the following in exponential forms

6. log636 = 2
7. log5125 = 3
8. log100.1 = –1
9. log4256 = 4
10. log981 = 2

II.     Answers to questions

6. log₆36 = 2
   6² = 36



7.  log₅125 = 3
   5³ = 125




8. log₁₀0.1 = –1
   10^-1 = 0.1




9. log₄256 = 4
   4⁴ = 256



10. log₉81 = 2
    9² = 81