Introduction to Adding and Subtracting Logarithms

Let a and b be any two positive real numbers. Then the formula for adding and subtracting them is given by:

 

(i) log a + log b = log (ab)

(ii) log a – log b = log `(a / b)`

(iii) log a (b) = `(log b) /( log a)`

 

Keeping these formulae in mind, we can solve many logarithmic problems.

Now let us see few problems on this topic adding and subtracting logarithms.

 

Example Problems on Adding and Subtracting Logarithms.

 

Ex 1: Solve log 2 (x) + log 4 (x) + log 16 (x) = `21 / 4`

Soln: Given: log 2 (x) + log 4 (x) + log 16 (x) = `21 / 4`

`=>` `log x / log 2` +` log x / log 4` + `log x / log 16` = `21 / 4`

`=>` log x `[1 / log 2 + 1 / log 2 ^2 + 1 / log 2 ^4]` = `21 / 4`

`=>` logx `[ 1 / log 2 + 1 / (2 log 2) + 1 / (4 log 2)]` = `21 / 4`

`=>` `log x / log 2` [ 1 +` 1 / 2` + `1 / 4` ] = `21 / 4`

`=>` ` log x / log 2` [ `1 3 / 4` ] = `21 / 4`

`=>`` log x / log 2` = `21 / 4 xx 4 / 7`

`=>` x = 2 ^3 `=>` x = 8.

 

Ex 2: Show that 7 log` (10 / 9)` – 2 log `(25 / 24)` + 3 log `(85 / 80)` = log 2

Soln: LHS  = 7 log `(10 / 9)` – 2 log `(25 / 24)` + 3 log `(81 / 80)`

                = 7 (log 10 – log 9) – 2 (log 25 – log 24) + 3 (log 81 – log 80)

               = 7 (log (2 `xx` 5) – log 2) – 2 log 5 2 + 2 log (3 `xx` 8) + 3 log 81 – 3 log 80

               = 7 log 2 + 7 log 5 – 14 log 3 – 4 log 5 + 2 log 3 + 6 log 2 + 12 log 3 – 3 (log (5 `xx ` 16))

               = 13 log 2 + 14 log 3 – 14 log 3 + 7 log 5 – 7 log 5 – 12 log 2

               = log 2 = RHS.

 

More Examples Example Problems on Adding and Subtracting Logarithms.

 

Ex 3: Solve: log x (8x – 4) – log (4) = 2

Soln: `log x ((8x ** 4) / 4 ) = 2`

`(8x ** 4) / 4` = x ^2 `=>` 2x – 1 = x 2

Therefore x 2 – 2x + 1 = 0  `=>` (x – 1) 2 = 0 `=>` x = 1.