Aptitude - Logarithm - Discussion

The logarithm of any number has two components. The characteristic and the mantissa.

Take for example, log103, the value of log103 = 0.4771.
Here, the 0 in the integral part is known as the characteristic and the value .4771 is known as the mantissa.

The value of log1030 is 1.4771.
Notice that the value of mantissa remained the same while that of the characteristic changed from 0 to 1.

Given the log of a number, we will be able to find out the number of digits that the original number had by knowing the value of the characteristic.

If the characteristic is '0', then the number is a single digit number
If the characteristic is '1', then the number is a two-digit number
If the characteristic is '5', then the number is a six-digit number

In the given problem, we need to find the number of digits of 3200.

If we take log we get log103200 = 200(log103) = 200 (0.4771) = 95.42.

Here, the characteristic is 95. Therefore, the number will have 96 digits.

The logarithm of a number contains two parts, namely 'characteristic' and 'mantissa'.

Characteristic: The internal part of the logarithm of a number is called its characteristic.

Case I: When the number is greater than 1.

In this case, the characteristic is one less than the number of digits in the left of the decimal point in the given number.

Case II: When the number is less than 1.

In this case, the characteristic is one more than the number of zeros between the decimal point and the first significant digit of the number and it is negative.

Instead of -1, -2 etc. we write 1 (one bar), 2 (two bar), etc.

For example 2^3=8 has 1 digit, but log 2^3 = 3*log2 = 3 (.301) = 0.903 so avoiding decimal part we get 0.

So, from given data, derived answer should be added with +1 to get original answer so finding digits use this.

Formula x^n= (log x^n) + 1 digits.

Let me solve another problem to understand easier:

10^2 = (log 10^2) + 1 = 2*log10 +1 =2 (1) +1 = 3 digits.

Now let us find digits in 2^64.

Log 2^64= (64*log 2) + 1 as->[ x^n = (log x^n) + 1 ].
=64* (0. 301) +1.
=20. 2.
Here don't confuse about the decimal part so let us take only 20.

Hence answer=20.

As given in important formulas when there are 'n' digits in the left-hand side of the decimal value of a logarithm the characteristic is added with 1 less than the no of digits.

So,
no. of digs =2(i.e.19) so we should add 1.
hence total = 19+1.
i.e. 20.

If there were 3 digits e.g. 345.8909: its characteristic will be "345" and number of digits =347 (i.e. 345+2).

Why 2? because no. of digs in characteristic = 3 and (3-1=2).

Simply we can calculate the number of digits in the given mathematical operation as below;

Here, the number of digits in 2^64 to be find then we have to calculate value of log 2^64.

So, log 2^64 = 19.26592.

And the result segregated as

19 is characteristic and 0.26592 is mantissa.

finally, we got 19 as characteristic so simply we can say the number of digits in 2^64 is 19+1=20.

Hence 20 is the correct answer.

If the logarithm to any base a gives characteristic n, then we say that the number of integers possible is given by:

a^(n+1)-a^n.

Here the characteristic is 19 and base is 10, therefore number of possible integers are:

10^(19+1) -10^19.
=>10^20-10^19.
=>10^19 (10-1).
=>9*10^19.

Thus it will have 19 zeroes and last digit 9 i.e.

90000000000000000000! count the digits you will get the answer :).

@All.

A logarithm consists of two parts:-.

1. The integral part called the characteristic and.
2. The decimal part called the mantissa.

It's a rule that to find a number of digits of any log, we just have to add 1 to the characteristic of the log.

For example, 2^64.

Log 2^64= 64*log 2.
=64* (0.301).
=19.2.

In this case, 19 is the characteristic.
So a number of digits is= 19+1.
=20.

Hence answer=20.

Characteristics is one less than the number of digits on the left of the decimal.

Hence, the answer we got 19 is characteristic since we used log. And it has 2 digits and char would be 1.

Now, the actual number of digits is 20. Why? because no. Of digits is one more than characteristics. Here the question actually doesn't have log so then we take the no. 19 and add char +1.

Hai.

If we elaborated 2^64, the result will be 18, 446, 744, 073, 709, 551, 616 which is exactly a 20 digit number.

But here the problem is that it will consume too much time for the calculation part.

It would be appreciative if someone tried to solve in a simple form.

Thank you.

There is actually no relationship between CHARACTERISTICS and DIGITS OF NUMBER.

Here log(2^64) = 64*log 2 = 64*0.3010 = 19.265.

Then, the equation of NUMBER OF DIGITS = NUMBER WHICH IS ON LEFT SIDE OF DECIMAL POINT + 1.

Therefore, here in this case -> 19+1=20.