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

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We use the relation for logarithm that states log a + log b = log (a*b).

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

=> E(x) = lg [ (1/8)*(9/10)*(10/11)*...*(999/1000)]

we see that we can cancel terms which are present in the denominator as well as numerator.

=> E(x) = lg ( (1*9)/(8*1000))

=> E(x) = lg (9 / 8000)

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