How do you calculate log0.000678?

Feb 14, 2018

$\log 0.000678 = - 4 + 0.83123$ or $\overline{4} .83123$.

Scientific calculators give it as $- 3.168773$

Explanation:

Here $\log$ means logarthim to the base $10$. When we take logarithm of a number, there are two parts of it; one part is characteristic and other mantissa.

While characteristic is the integral part and mantissa is the fractional or decimal part. For example $\log 500 = 2.6990$. Here $2$ is characteristic and $0.6990$ is mantissa.

While characteristic can be any integer, mantissa cannot be a negative number and is always positive. For example we write $- 2.3010$ as $- 3 + 0.6990$ i.e. a sum of an integer and a positive proper fraction and here for $- 2.3010$, charcteristic is $- 3$ and mantissa is $0.6990$.

Characteristic depends on the place from where the number starts. For example, for a three digit number like $523$ it is $2$, for a six digit number $743892$ it is $5$. If we have a number $8.375$, characteristic is $0$.

What about numbers less than $1$, such as $0.893$ or $0.00893$ or $0.00000893$. In such cases characteristic is negative and depends on the place from where the number starts. For $0.893$ characteristic is $- 1$; for $0.00893$ it is $- 3$ and for $0.00000893$ it is $- 7$.

Mantissa on the other hand is independent of the position of the decimal point in the number and just depends on first four digits, excluding $0 ' s$ on the left and is given in the logarithmic tables.

Hence as in $\log 0.000678$, number starts from fourth place after decimal, characteristic is $- 4$ and tables give mantissa as $83123$ (they are easily available on web) and hence

$\log 0.000678 = - 4 + 0.83123$ and is also written as $\overline{4} .83123$.

Scientific calculators give it as $- 3.168773$