Question

Show that, 7 log⁡〖(10/9〗)+2 log⁡〖(81/80〗)=2 log⁡〖(25/24〗)+log⁡2

Answer 1

Step-by-step explanation:

Given:-

7 log (10/9) + 3 log (81/80)

To find:-

Show that:-

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

Solution:-

LHS:-

7 log (10/9) + 3 log (81/80)

We know that

log (a/b) = log a - log b

=> 7(log 10 - log 9 ) + 3( log 81- log 80)

=>7 log 10 - 7 log 9 + 3 log 81 - 3 log 80

=> 7 log (2×5) - 7 log 3^2 + 3 log 3^4

- 3 log (2^4×5)

We know that log (ab) = log a + log b

=>7 log 2+ 7 log 5 -7 log 3^2+3 log 3^4

-3 log 2^4+ 3 log 5

We know that

log a^m = m log a

=>7 log 2 + 7 log 5 -(2×7) log 3 +

(3×4) log3 -(3×4) log 2 + 3 log 5

=> 7 log 2 + 7 log 5 -14 log 3 + 12 log 3

-12 log 2 -3 log 5

=> (7-3) log 5 +(-14+12) log 3+(7-12 ) log 2

=> 4 log 5 -2 log 3 -5 log 2 -------------(1)

RHS:-

2 log (25/24) + log 2

We know that

log (a/b) = log a - log b

=> 2 log 25 - 2 log 24 + log 2

=> 2 log 5^2 - 2 log (2^3 ×3) + log 2

We know that

log (ab) = log a + log b

=> 2 log 5^2 - 2 log 2^3 - 2 log 3+ log 2

We know that

log a^m = m log a

=> (2×2) log 5 -(2×3) log 2 - 2 log 3 + log 2

=> 4 log 5 -6 log 2 - 2 log 3 + log 2

=> 4 log 5 -5 log 2 -2 log 3

=> 4 log 5 - 2 log 3 - 5 log 2 --------------(2)

From (1) &(2)

LHS = RHS

Answer:-

7log(10/9)+3log(81/80)=2log(25/24)+log2

Used formulae:-

• log (a/b) = log a - log b

• log (ab) = log a + log b

• log a^m = m log a