Ratios, Logarithms, and Decibels (things to remember)

When appearing for competitive examinations or elsewhere, it is important to find solutions quickly and near accurately to mathematical problem involving logarithms (especially when Log Table is not available). In such situation, the problem involving logarithm can be easily and quickly solved almost accurately by remembering the logs of 1, 2, 3, 7 and 10 with the knowledge of ‘Laws of Logarithms’.

log of 1 = 0
log of 2 = 0.3
log of 3 = 0.48
log of 7 = 0.85
log of 10 = 1

Examples

1. log of 4 = ?

We know from the Laws of logarithm that,

log (m*n) = log (m) + log (n)

Since we know the log of 2 is 0.3 and 4 is a multiple of 2. In our example, 4 can be rewritten as 2 multiplied by 2 i.e., 4 = 2*2. Hence, log 4 can be rewritten as log (2*2).

log (4) = log (2*2) = log (2) + log (2);
log (4) = 0.3 + 0.3;
log (4) = 0.6 ; From log table, log (4) = 0.6020

2. log of 5 = ?

We know from the Laws of logarithm that,

log (m/n) = log (m) - log (n)

In this example, 5 can be rewritten as 10 divided by 2 i.e., 5 = 10/2. Hence, log (5) can be rewritten as log (10/2).

log (5) = log (10/2) = log (10) – log (2);
log (5) = 1 – 0.3;

log (5) = 0.7 ; From log table, log (5) = 0.6989