Relationship between Indices & Logarithms

Recall the laws of indices;

These laws are true for all values of x, y and a ≠0. It is common knowledge that scientific calculators have taken the place of logarithm and anti-logarithm tables. However, it is important to note that the theory of Logarithms is very relevant in science and technology. 

Note that in index notation 3⁴ = 81

Where 81 is the number, 3 is the base and 4 is the index or power.

But it can also be said that the logarithm of 81 to base 3 is 4.

i.e.   81 = 3⁴      ……..(1)

and log₃81 = 4      ………(2)

It is important to note that equations (1) and (2) are equivalent.

3⁴  = 81 is the index form while log₃81 = 4 is the logarithmic form.

It is necessary to understand how to change from one form to the other. The general form is given as:

If  N = a^x, then log_aN = x

So we can have the following:

Example 5.1.1:

Evaluate the following logarithms without using tables or a calculator:

1. log₂8                    
2. log₃81              
3. log₃729              
4. log_0.25128
5. log₈0.0625              
6. log_1.21.728          
7. \( \scriptsize \log_{\sqrt [4]{ 2}} \sqrt{512} \)          
8. \( \scriptsize \log_{\sqrt [2]{2}}\sqrt{128} \)

Solution:

1. log₂8  

Let log₂8 = x