# logarithmic functions

## logarithmic Rules & Formula

The logarithm of a given number b to the base ‘a’ is the exponent indicating the power to which the base ‘a’ must be raised to obtain the number b. This number is designated as log a b. Hence log a b = x ⇔ ax = b, a > 0, a 1 and b>0. From the definition of the logarithm of the number b to the base ‘a’, we have an identity This is known as Fundamental Logarithmic Identity.

### GRAPH OF LOGRITHMIC FUNCTION ### PROPERTIES & Formulas OF LOGARITHMIC FUNCTION

• The expression is meaningful for b >0 and for either 0 < a < 1 or a > 1.

• Let a > 1, then and

If 0 < a < 1, then • log a(mn) = log a m + log a n

• , c > 0 and c ≠ 1.

• log a = log a m – log a n

• log a mn = n log a m

• provided both a and b are non-unity.

• loga1 = 0

• logaa = 1

• Example: Solve for x: log ½ (x – 2) > 4

Detail Explanation :  In such type of questions first we make the base same.

Given that log ½ (x – 2) > 4 log ½ ½

log ½ (x – 2) > log ½ 1/16

x – 2 < 1 / 16

x < 33/16

also x – 2 > 0 ⇒ x > 2

hence x ∈ (2, 33/16)

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