# 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 10^{2} = 100, the logarithm of 100 to the base 10 is 2.

This is written as:

log_{10} 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} (m^{n}) = 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^{loga 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.