Aptitude - Logarithm - Discussion

@Pawan here it's given 2^64. You can find the factor of 64 and then calculations will be lengthy. So the given solution is accurate.

Why +1 ?

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.

When We Are Finding Number of Digits We Have To Always Add +1 to The Characteristics Value. So, 19 + 1 =20.

2^64=10^x.
X=log 2^64.
X=19.164.

2^64=10^(19.164)=10^(19+.164)=1.8*10^19=18*10^18.
Hence after 18, There are 18 digits(zeros).
Total digits = 2(i.e. 1,8)+18(zeros) = 20.

@Vicky.

The value of log 103 is 0.4771 but here are three digits, 1 0 3. How can you say that there is a single digit?

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.

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.

@Gopi.

Value of log (103) is 2.01 so number Of digits are 2+1.

Thanks for explaining @Vicky.