Logarithms of Numbers less than 1 (negative characteristics) - SS2 Mathematics Lesson Note

The characteristic of the logarithm of numbers between 0 and 1 is negative. Usually, when a number is expressed in standard form as \(A \times 10^{n}\) or \(A \times 10^{- n}\), the exponents \(n\) and \(–n\) compose the characteristic of the log of those numbers.

For numbers less than 1, the characteristic of the log of said numbers is obtained by adding 1 to the number of zeroes between the decimal point and the first non-zero significant digit; then make the number negative. Consider, the number \(0.0314\), there is one zero between the decimal point and \(3\) (the first significant digit). So we add 1 and \(1 + 1 = 2\). Making this negative, we obtain \(\overline{2}\), read as “bar two” not “minus two”.

So the characteristic of the log of a number can either be positive, negative or zero, whilst the mantissa will always be positive. The mantissa is numbers less than 1 is found the same way as numbers greater than 1.

Example 4 Find the logarithm of the following:

1. \(0.051\)

2. \(0.0000765\)

3. \(0.238\)

Solution

1. \(\log{0.051}:\)

   1. The characteristic = \(number\ of\ zeroes\ b/w\ the\ decimal\ point\ and\ the\ 1st\ significant\ digit\)

\[= 1 + 1 = 2 = \overline{2}\]

1. The mantissa is \(51 = 7070\)

2. \(\log{0.051} = \ \overline{2}.7076\)

1. \(\log{0.0000765}:\)

   1. The characteristic = \(number\ of\ zeroes\ b/w\ the\ decimal\ point\ and\ the\ 1st\ significant\ digit\)

\[= 4 + 1 = 5 = \overline{5}\]

1. The mantissa is \(76\) under \(5\)=\(8837\)

2. \(\log{0.051} = \ \overline{5}.8837\)

1. \(\log{0.238}:\)

   1. The characteristic = \(number\ of\ zeroes\ b/w\ the\ decimal\ point\ and\ the\ 1st\ significant\ digit\)

\[= 0 + 1 = 1 = \overline{1}\]

1. The mantissa is \(23\) under \(8\) = \(3766\)

2. \(\log{0.051} = \ \overline{1}.3766\)