# Log(0.0058) is -2.23657, integral part is -2 but not -3

prashant singh
Log(0.0058) is ( -2.2365720064), its characteristic or integral part is (-2) but not (-3). As per rules of logarithm, Its characteristic or integral part must be (-3 ) because of two zeros plus 1 (as per rule) but its characteristic is (-2), similiarly log(0.0648) = -1.188424249941 but integral part must be (-2) but here it is (-1), I know why it is so but I only want to know why rule is getting wrong, if rule is wrong then why they made it. Please answer

NickTesla

Staff Emeritus
Homework Helper
Log(0.0058) is ( -2.2365720064), its characteristic or integral part is (-2) but not (-3). As per rules of logarithm, Its characteristic or integral part must be (-3 ) because of two zeros plus 1 (as per rule) but its characteristic is (-2), similiarly log(0.0648) = -1.188424249941 but integral part must be (-2) but here it is (-1), I know why it is so but I only want to know why rule is getting wrong, if rule is wrong then why they made it. Please answer
You're forgetting that while the characteristic is negative, the basic numerical part of the decimal has a positive logarithm.

For example, taking x = 0.0058, one would calculate the characteristic by moving the decimal three places to the right of its original location, which means the characteristic is -3 as you say. But after you do this, the original number is converted from 0.0058 to 5.8. The number 5.8 has a log of about 0.763428, which must be added algebraically to the characteristic, in keeping with the laws of logarithms.

Therefore x = 0.0058 = 5.8 × 10-3.

Taking logs: log (x) = log [5.8 × 10-3] = log (5.8) + log (10-3) = 0.763428 + (-3) = -2.236572

When using log tables, the log (x) for x < 1 can be written using bar-notation, which is explained in this
article:

https://en.wikipedia.org/wiki/Common_logarithm

The bar-notation helps when going back and forth between numbers and log tables, which one does not need to do with a calculator.

prashant singh
Wow great sir , u mean that the value which we get from log tables are not correct value unless we remove the dot sign between characteristic and mantisa with the plus sign, u mean that for example log(0.0000048) = 6.68124 , this is a log table answer but it is not correct until we write it as -6 + o.68124 = -5.31875, original answer which matches with calculator. You mean that characteristic means from where we can start calculating logarithm and therefore it has bar symbol otherwise it may have only negative sign

Staff Emeritus
Homework Helper
Wow great sir , u mean that the value which we get from log tables are not correct value unless we remove the dot sign between characteristic and mantisa with the plus sign, u mean that for example log(0.0000048) = 6.68124 , this is a log table answer but it is not correct until we write it as -6 + o.68124 = -5.31875, original answer which matches with calculator. You mean that characteristic means from where we can start calculating logarithm and therefore it has bar symbol otherwise it may have only negative sign

All I'm saying is that the log of a number less than 1 can be written in two different ways. If you use a calculator to calculate said log, you will get only one version of the log (-5.31875 to use the example above), not both.

The other version of the log, the so-called bar format (##\bar 6.68124##), can be used with log tables to calculate the logarithm in that manner, but you won't get it using a calculator to calculate the logarithm.

prashant singh
prashant singh
Yes sir I got it , thanks for your great help, I love u
All I'm saying is that the log of a number less than 1 can be written in two different ways. If you use a calculator to calculate said log, you will get only one version of the log (-5.31875 to use the example above), not both.

The other version of the log, the so-called bar format (##\bar 6.68124##), can be used with log tables to calculate the logarithm in that manner, but you won't get it using a calculator to calculate the logarithm.

prashant singh
Sir directly using the bar method in a question will be wrong or right , because bar method of finding the logarithm is not the correct value we have to simplify it and get the correct value

Staff Emeritus