Question

prove 7 log 16/15 +5 log 25/24 +3log 81/80 =log 2

Answer 1

The given expression = log (16/15)^7 + log (25/24)^5 + log (81/80)^3 using the rule n log m = log (m^n)

= log {2^4/(3)(5)}^7 + log { 5^2/(2^3 .3)}^5 + log {3^4/2^4.5)}^3 expressing the composite numbers in terms of product of prime numbers.

= log {2^28/3^7.5^7} + log {5^10/2^15.3^5} + log {3^12/2^12.5^3} applying the power law of indices.

= log {(2^28 x 5^10 x 3^12)/(3^7 x 5^7 x 2^15 x 3^5 x 2^12 x 5^3)} using the rule log a + log b + log c = log abc

= log {(2^28 x 3^12 x 5^12)/(2^27 x 3^12 x 5^10)}

= log 2.