We'll use the product property of logarithmic function, which means that the sum of logarithms is the logarithm of product, we'll get:
lgx1+lgx2+lgx3+....+lgxn=lg(x1*x2*x3*...*xn)
In our case:
lg(1/2)+lg(2/3)+...+lg(99/100)=
lg((1/2)*(2/3)*(3/4)*...*(98/99)*(99/100))=lg(1/100)=-2
Now, we'll use the quotient property:
lg(1/100)=lg1 - lg100 = 0 - lg(10)^2 = 0-2lg10 = 0-2 = -2
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