LOGARITHM

If a is a positive real number other than 1, and a^m = x, then we can write: m = log_ax, 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

log_10⁵  = 0.6989 ,       log_{10^{( 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.

Quant Lessons :
Counting In Geometry!
last non zero digit of factorial -TG Math Corner