**Rules for Logarithms**

The following are the rules for logarithms are nothing but power laws of exponents which are studied in earlier classes. These rules hold for any base 'a' (a > 0 and a ≠ 1 )## Rules for logarithms

**1st rule : log**

_{a}(mn) = log_{a}m + log_{a}n**Proof :**Suppose log

_{a}m = x and log

_{a}n = y

a

^{x}= m and a

^{y}= n

∴ mn = a

^{x}. a

^{y}

mn = a

^{x + y}

By definition of logarithm

log

_{a}(mn) = x + y

log

_{a}(mn) = log

_{a}m + log

_{a}n

**Example :**Prove that : log 12 = log 3 + log 4

**Solution :**Consider log 3 + log 4

log (3 x 4) --------> By 1st rule of logarithms

= log 12

∴ log 12 = log 3 + log 4

**2nd rule : log**

_{a}(m/n) = log_{a}m - log_{a}n**Proof :**Suppose log

_{a}m = x and log

_{a}n = y

a

^{x}= m and a

^{y}= n

∴ m/n = a

^{x}/ a

^{y}

m/n = a

^{x - y}

By definition of logarithm

log

_{a}(m/n) = x - y

log

_{a}(m/n) = log

_{a}m -log

_{a}n

**Example :**Find log

_{2}16 - log

_{2}8

**Solution :**Consider log

_{2}16 - log

_{2}8

log

_{2}16 - log

_{2}8 = log

_{2}(16/8) --------> By 2nd rule of logarithms

= log

_{2}2

= 1 ( By properties of logarithm)

**3rd rule : log**

_{a}(m)^{n}= n log_{a}m**Proof :**Suppose log

_{a}m = x

a

^{x}= m

(a

^{x})

^{n}= (m)

^{n}( taking nth power on both sides)

By definition of logarithm

log

_{a}(m)

^{n}= n. x

log

_{a}(m)

^{n}= n log

_{a}m

**Example :**Find log

_{6}36

**Solution :**log

_{6}36

= log

_{6}6

^{2}

= 2 log

_{6}6 ( By 3rd rule)

∴ log

_{6}36 = 2

**Solve for x :**

1) x = log

_{7}343

x = log

_{7}7

^{3}

x = 3 log

_{7}7 (by 3rd rule)

x = 3 x 1

x = 3

2) log 2 + log(x+2) - log (3x-5) = log 3

**Solution :**log 2 + log(x+2) - log (3x-5) = log 3

By 1st rule

log 2(x+2) - log (3x-5) = log 3

By 2nd rule log 2(x+2)/(3x-5) = log 3

2(x+2)/(3x-5) = 3/1

2(x+2) = 3(3x-5) -----(cross multiply)

2x + 4 = 9x -15

2x -9x = -15-4

-7x = -19

∴ x = 19/7

From Rules of logarithms to Home Page