Question

Find the value of \[\log 0.000678\].

Answer 1

Hint: Log means logarithm which is a power raised to the base to yield another number. For example, the log $100$ of base $10$ is\[2\]. The logarithm of a number has two parts, one is characteristic and another one is mantissa. For example, $\log 453 = 2.65609$, here, the characteristic is 2 and the mantissa is 65609.

Complete step-by-step solution:
In this problem, we have to find the value of \[\log 0.000678\]. The characteristic of a logarithm is found by the place from where the non-zero digit starts. For example, the characteristic of a three digit number $(234)$ is $2$, for a six-digit number $(456738)$ it is 5 but it we have a number like $6.897$ then the characteristic is 0 and for the numbers that are less than $1$, then the characteristic of that number is negative.

For $\log 0.003459$ the characteristic is $ - 3$ , as the non-zero digit starts from the third place of the decimal, for $\log 0.876$ characteristic is $ - 1$, as the non-zero digit starts from first place. The mantissa part of logarithm depends on the first four digits of the number, from where the non-zero digit starts and it doesn’t depend on the position of the decimal in the number. The mantissa part is calculated with the help of a log table.

Hence, the characteristic part of \[\log 0.000678\] is $ - 4$, as the non-zero digits start from the fourth place and the mantissa is 83123, found from the log table. And we know that this mantissa part is in decimal. The log is the sum of characteristic and the mantissa part. So, the value of \[\log 0.000678\] is$ - 4 + 0.83123 = - 3.16877$


Note: Characteristic can be positive or negative but the mantissa part can never be negative, it remains always positive. The integral part is the characteristic and the fractional or decimal part is the mantissa. The value of \[\log 0.000678\] is also written as $\bar 4.83123$.