Evaluate expression with help of properties of logarithms E(x)= lg(1/8)+lg(9/10)+lg(10/11)+...+lg(999/1000)

To evaluate expression with help of properties of
logarithms:

E(x)=
lg(1/8)+lg(9/10)+lg(10/11)+...+lg(999/1000).

Specially it
is  noticed  that there is no term lg(8/9) on the right 
side.

Solution:We use the property of logarithms: lga+lgb =
lgab.

Therefore E(x) =
lg(1/8)+lg(9/10)+(10/11)+lg(11/12)...+lg(998/999)+lg(999/1000).

E(x)
= lg(1/8)+lg(9/10)(10/11)(11/12)(12/13)....(998/999)(999/1000)}, as lga+lgb =
lgab.

E(x) = lg(1/8) + lg{9/1000) as other terms
cancel.

So E(x) = lg{(9/8000), by property lga*lgb = lg
ab.

So E(x) = lg9-log8
-lg1000

So E(x) = lg9-lg8 -
3.

If there was the 2nd term  lg(8/9), the sum E(x) =
lg(8/9)+lg(9)-lg8-3 = lg8-lg9+lg8-lg9 3 =
-3.

So E(x) = lg9-lg8 - 3 =
-2.948847478.