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), a^m = b, then the index m is called the logarithm of b to the base a.

b = a^m

log b = log a^m

log b = m log a

(log b)/(log a) = m

log_a b = m

Therefore, if b = a^m, then m = log_a b

Laws of Logarithmic Identities

1. Product formula

The logarithm of the product of two numbers is equal to the sum of their logarithms.

log_a (mn) = log_a m + log_a n

2. Quotient formula

The logarithm of the quotient of two numbers is equal to the difference of their logarithms.

log_a (m/n) = log_a m - log_a n

3. Power formula

The logarithm of a number raised to a power is equal to the power multiplied by logarithm of the number.

log_a (mⁿ) = n log_a m

4. Base changing formula

log_n m = (log_a m)/(log_a n)

5. Reciprocal relation

log_b a × log_a b = 1

6. log_b a = 1/(log_a b)

7. a^{log_a x} = x

8. If a > 1 and x > 1, then log_a x > 0.

9. If 0 < a < 1 and 0 < x < 1, then log_a x > 0.

10. If 0 < a < 1 and x > 1, then log_a x > 0.

11. If a > 1 and 0 < x < 1, then log_a x < 0.