Logarithms

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

Laws of Logarithmic Identities

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.