(lg5)² + lg2 lg50

= (lg5)² + lg2 lg(5² x 2)

= (lg5)² + lg2 (lg5² + lg2)

= (lg5)² + lg2 (2lg5 + lg2)

= (lg5)² + 2 lg5 lg2 + (lg2)²

= (lg 5 + lg2)²

= (lg(5 x 2))²

= (lg10)²

= 1²

= 1

lg (1/2) + lg (2/3) + lg (3/4) +... + lg (998/999) + lg (999/1000)

= lg 1 - lg 2 + lg 2 - lg 3 + lg 3 - lg 4 + ... + lg 998 - lg 999 + lg 999 - lg 1000

= lg 1 - lg 1000
(notice that the above terms cancel out except these two)

= 0 - lg10³

= 0 - 3 lg10

= 0 - 3

= -3

log√2 (√9 - √5)

= log√2 ( (√9 - √5) x (√9 + √5)/(√9 + √5) )

= log√2 ( ((√9)² - (√5)²) / (√9 + √5) )

= log√2 ( (9 - 5) / (√9 + √5) )

= log√2 ( 4 / (√9 + √5) )

= log√2 (4) - log√2 (√9 + √5)

= log√2 (√2)⁴ - ( log2 (√9 + √5) / log2 (√2) )

= 4 log√2 (√2) - ( k / log2 (2)¹⁄² )

= 4 - k / (½ log2 (2) )

= 4 - k / ½

= 4 - 2k

Based on the identity given,

u4 = (1 - ¼)²

u4 = (¾)²

lg u4 = lg (¾)²

= 2 lg ¾

u(10^k) = (1 - 1/10^k )²

lg u(10^k) = lg (1 - 1/10^k )²

lg u(10^k) = 2 lg (1 - 1/10^k )

lg u(10^k) = 2 lg ( (10^k - 1)/10^k )

You can skip the first 3 steps if you are able to see the pattern directly (the numerator is always 1 smaller than the denominator)

lg u2 + lg u3 + lg u4 + ... lg u(10^k)

= 2 lg (1/2) + 2 lg (2/3) + 2 lg (3/4) + ... 2 lg ( (10^k - 1)/10^k )

= 2lg1 - 2lg2 + 2lg2 - 2lg3 + 2lg3 - 2lg4 + ... - 2lg (10^k - 1) + 2lg (10^k - 1) - 2lg (10^k)

= 2lg1 - 2lg (10^k)

(Common terms cancel out again, like in the first part of the question

= 2(0) - 2(k lg10)

= 0 - 2k

= -2k

Thank you!