Question

prove that : 2log 15/18 - log 25/162+ log 4/9 = log 2

Answer 1

hello friends..

Formula used:
log mn = log m + log n
log m/n = log m- log n

hence :
2log 15/18 - log 25/162+ log 4/9 = log 2

hope it helps...

Answer 2

LHS = 2㏒ $\frac{15}{18}$ - ㏒ $\frac{25}{162}$ + ㏒ $\frac{4}{9}$

= ㏒ $( \frac{15}{18})^{2}$ - ㏒ $\frac{25}{162}$ + ㏒ $\frac{4}{9}$

= ㏒ $\frac{225}{324}$ - ㏒ $\frac{25}{162}$ + ㏒ $\frac{4}{9}$

= ㏒ 225 - ㏒ 324 - [ ㏒ 25 - ㏒ 162 ] + ㏒ 4 - ㏒ 9

= ㏒ 225 - ㏒ 324 - ㏒ 25 + ㏒ 162 + ㏒ 4 - ㏒ 9

= ㏒ 15² - ㏒ 324 - ㏒ 25 + ㏒ 162 + ㏒ 2² - ㏒3²

= 2 ㏒ (3×5) - ㏒ (2²×3⁴) - ㏒ 5² + ㏒ (2×3⁴) + 2㏒2 - 2㏒3

= 2㏒3 + 2㏒5 - [㏒ 2² + ㏒ 3⁴] - 2㏒5 + ㏒ 2 + ㏒ 3⁴ + 2 ㏒2 - 2㏒3

= - 2㏒2 - 4㏒3 + ㏒ 2 + 4㏒3 + 2㏒2

= ㏒ 2

= RHS

Hence proved...

Hope it helps