## Number of Digits in the Exponent of a Number

**LOGARITHM**

If *a* is a positive real number other than 1, and *a ^{m}* =

*x*, then we can write:

*m = log*and we say that the value of log

_{a}x,*x*to the base

*a*is

*m*.

Examples:

(i) 10^{4} = 10000 log_{10} 10000 = 4.

(ii) 3^{5} = 243 log_{3} 243 = 5.

(iii) (0.1)^{3} = 0.001 log_{ (0.1)} (0.001) = 3.

Properties of Logarithms

**Number of digits in the exponent of a number :**

Look at these numbers

log_{10}^{2} = 0.3010 , log_{10}^{( 2 × 10)} = log_{10}^{2} + log_{10}^{10} = 1. 3010 , log_{10} ^{(2 × 102)} = 2 . 3010 , log_{10}^{(2 × 103)} ^{ } = 3.3010

log_{10}^{3} = 0.4771, log_{10}^{(3×10)} = log_{10}^{3} + log_{10} ^{10} = 1.4771, log_{10}^{ (3×102)} = 2.4771, log_{10}^{(3×103)} = 3.4771

log_{105 } = 0.6989 , log_{10( 5 × 10)} = 1. 6989 , log_{10} ^{(5 × 102)} = 2. 6989 etc.

**Thus in base 10, the integral part of the logarithm represents the power of 10 in the number.**

Also,

log_{10} ^{(2/10)} = log_{10}^{2} – log_{10}^{10} = 0.3010 – 1, log_{10} ^{(2/100)} = 0.3010 – 2 , log_{10} ^{(2/1000)}=0.3010 – 3

log_{10} ^{(3/10)} = log_{10}^{3} – log_{10}^{10} = 0.4771 – 1, log_{10} ^{(3/100)} = 0.4771 – 2, log_{10} ^{(3/1000)} = 0.4771 – 3 etc.

Thus, if the integral part is negative, it represents the negative power of 10 in the number (provided that the fractional part of the logarithm is positive).

Let us take two numbers 2^{100} and (1/2)^{100}, the logarithm of the first number is 100log_{10}2 = 30.10, it means the number 2^{100} has 30 powers of 10 multiplied by a single digit number, thus it has 31 digits. (That’s why we add 1 to this number to calculate the number of digits).

The logarithm of the second number is -100log_{10}2 = – 30.10= -31 + 0.90. This means that the number (1/2)^{100} has non-zero digit at the 31st place after the decimal. (Remember that 10^{-3} has a non-zero digit at the 3rd place after the decimal). Thus, there are 30 zeros between the decimal point and the first non-zero digit in (1/2)^{100}.

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