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### Problem Statement

The “multiplier of the variance ” function grows very slowly. To understand how very slowly, calculate a table with n = 10j and for j = 10, 20, 30,..., 100. How large must n be for to exceed 10?, to exceed 20? (Remember, in mathematical work above calculus, log(x) is the natural logarithm, base e, often written ln(x) in calculus and below to distinguish it from the “common” or base-10 logarithm. Be careful, some software and technology may not properly calculate with magnitudes this large.)

### Solution

 j n = 10j 10 1.0E+10 2.50464270 20 1.0E+20 2.76758549 30 1.0E+30 2.91040541 40 1.0E+40 3.00762760 50 1.0E+50 3.08092695 60 1.0E+60 3.13954678 70 1.0E+70 3.18826840 80 1.0E+80 3.22987897 90 1.0E+90 3.26614210 100 1.0E+100 3.29824275

For to exceed 10, n must be larger than ee50 e5.18×1021. This is a number which would have about 2.24 × 1021 decimal digits.

For to exceed 20, n must be larger than ee50 e7.23×1087. This is a number which would have about 3.13 × 1087 decimal digits.

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