# LOGARITHM

If a is a positive real number other than 1, and am = x, then we can write: m = logax, and we say that the value of log x to the base a is m.
Examples:

(i) 104 = 10000     log10 10000 = 4.

(ii) 35 = 243     log3 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

log102 = 0.3010 ,        log10( 2 × 10) = log102 + log1010 = 1. 3010 ,      log10 (2 × 102) = 2 . 3010 ,   log10(2 × 103)   = 3.3010

log103 = 0.4771,         log10(3×10) = log103 + log10 10 = 1.4771,         log10 (3×102) = 2.4771,         log10(3×103) = 3.4771

log105  = 0.6989 ,       log10( 5 × 10)  = 1. 6989 ,                                    log10 (5 × 102) = 2. 6989 etc.

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

Also,
log10 (2/10) = log102 – log1010 = 0.3010 – 1,        log10 (2/100) = 0.3010 – 2 ,         log10 (2/1000)=0.3010 – 3

log10 (3/10) =  log103 – log1010 = 0.4771 – 1,      log10 (3/100) = 0.4771 – 2,            log10 (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 2100 and (1/2)100, the logarithm of the first number is 100log102 = 30.10, it means the number 2100 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 -100log102 = – 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.