How many digits does 125100 have?
Solve it without using values such as log102 or log105.
Step 1 · Rewrite
125100 = 5300
Because 125 = 5³.
Step 2 · Create powers of 10
5300 = 10300 / 2300
Since 2³⁰⁰·5³⁰⁰ = 10³⁰⁰.
Step 3 · Bound 2³⁰⁰
1090 < 2300 < 1091
This is the only non-obvious part.
Why is that bound true?
Use the convenient exact power
230 = 1,073,741,824.
109 < 230 < 1.1 × 109
↓ raise to the 10th power
1090 < 2300 <
1.110 × 1090
Since 1.110 ≈ 2.59 < 10,
1090 < 2300 < 1091.
Finish
1090 < 2300 < 1091
↓ divide 10³⁰⁰ by the bounds
10209 <
10300/2300
< 10210
10209 < 125100 < 10210
Therefore, 125100 has 210 digits.
Interview intuition
When the base contains powers of 5, pair them with powers of 2 to manufacture powers of 10. Then estimate the remaining power using a convenient exact benchmark such as 230.