x->18->1:(两处红色标记是我个人加上的，怀疑原文有误，即若有 10 和 100，则前面不应有 90 和 1800)

x=log2+log3+...+log9
+90+log1.1+log1.2+...+log9.9
+1800+log1.01+log1.02+...+log9.99
+3
=∫logx dx (从2到10)
+90+10∫logx dx(从1.1到9.9)
+1800+ 100∫logx dx (从1.01到9.99)
+3
= ...

x->18->2:
x=(∫log(x)dx(2--1001)+∫log(x)dx(1--1000))/2
=((x*log(x)-∫xdlog(x))(2--1001)+(x*log(x)-∫xdlog(x))(1---1000))/2
=2567.857000.....

• length(123456789)10=floor[lg(123456789)+1]=floor[8.091514977+1 ]=9
• length(100000000)10=floor[lg(100000000)+1]=floor[8+1]=9
• length(10101)2=floor[log 2 (23) + 1]=floor[4.523561956+1]=5  (10101)2=(23)10

length(N)10=floor[lg(N)+1]
=floor[lg(1*2*3*...*999*1000)+1]
=floor[lg1+lg2+lg3+...+lg999+lg1000+1]
=floor[lg2+lg3+...lg999+lg1000+1]    <= lg1=0

∑(N=2..1000)lgN = ∫lgxdx (x=2..1000)

∫lgxdx (x=2..1000) = (1/ln10)*∫lnxdx (x=2..1000)
= (1/ln10)*[x*lnx - ∫xd(lnx)] (x=2..1000)
= (1/ln10)*[x*lnx - ∫dx] (x=2..1000)
= (1/ln10)*[x*lnx - x] (x=2..1000)
= x*(lnx - 1)/ln10 (x=2..1000)

∫lgxdx (x=2..1000) = 1000*(ln1000 - 1)/ln10 - 2*(ln2 - 1)/ln10
= [1000*(6.907755279 - 1) - 2*(0.693147181 - 1)]/ln10
= [1000* 5.907755279 - 2*(-0.306852819)]/2.302585093
= [5907.755279 - (- 0.613705639)]/2.302585093
= 5908.368984639/2.302585093
= 2565.97204707

length(N)10 = floor[2565.97204707 + 1] = 2566

∫lgxdx (x=2..1000) = 1000*(ln1000 - 1)/ln10 - 1*(ln1 - 1)/ln10
= [1000*( 6.907755279 - 1) - 1*(0 - 1)]/ln10
= [1000*5.907755279 - 1*(-1)]/2.302585093
= [5907.755279 + 1]/2.302585093
= 5908.755279/2.302585093
= 2566.13981258

length(N)10 = floor[2566.13981258 + 1] = 2567