Logarithm is the exponent or power to which a stated number called the base is raised to yield a specific number. For example, in the expression 102 = 100, the logarithm of 100 to the base 10 is 2.
This is written as:
log10 100 = 2
If for a positive real number (a ≠ 1), am = b, then the index m is called the logarithm of b to the base a.
b = am
log b = log am
log b = m log a
(log b)/(log a) = m
loga b = m
Therefore, if b = am, then m = loga b
1. Product formula
The logarithm of the product of two numbers is equal to the sum of their logarithms.
loga (mn) = loga m + loga n
2. Quotient formula
The logarithm of the quotient of two numbers is equal to the difference of their logarithms.
loga (m/n) = loga m - loga n
3. Power formula
The logarithm of a number raised to a power is equal to the power multiplied by logarithm of the number.
loga (mn) = n loga m
4. Base changing formula
logn m = (loga m)/(loga n)
5. Reciprocal relation
logb a × loga b = 1
6. logb a = 1/(loga b)
7. aloga x = x
8. If a > 1 and x > 1, then loga x > 0.
9. If 0 < a < 1 and 0 < x < 1, then loga x > 0.
10. If 0 < a < 1 and x > 1, then loga x > 0.
11. If a > 1 and 0 < x < 1, then loga x < 0.