Logarithms • Topic 4 of 4

Characteristic & Mantissa

Every base-10 logarithm splits into two parts: the characteristic (the integer part) and the mantissa (the non-negative decimal part). For a number N greater than 1, the characteristic of log_10 N is exactly one less than the number of digits — so digits = (characteristic of log N) + 1 = ⌊log_10 N⌋ + 1. That is the CAT digit-counting trick: to find how many digits 2^100 has, compute 100·log 2 = 100 × 0.30103 = 30.103, take the floor (30), add 1 → 31 digits. The mantissa (here 0.103) fixes the leading digits but you rarely need it for digit counts. For a positive number less than 1, the characteristic is negative and tells you the position of the first significant figure after the decimal point: if log N = −2.7, there are 1 leading zeros after the decimal before the first significant digit. Keep log 2 ≈ 0.30103 and log 3 ≈ 0.47712 handy — almost every CAT digit question uses them.

✅ Solved examples

1. How many digits are in 2^50? (log 2 = 0.30103)
log(2^50) = 50 × 0.30103 = 15.0515. Characteristic 15 ⇒ digits = 15 + 1 = 16.
2. How many digits does 6^20 have? (log 2 = 0.30103, log 3 = 0.47712)
log(6^20) = 20(log 2 + log 3) = 20 × 0.77815 = 15.563. Floor 15 ⇒ 16 digits.
3. Find the number of digits in 5^100. (log 2 = 0.30103)
log 5 = log(10/2) = 1 − 0.30103 = 0.69897. ×100 = 69.897. Floor 69 ⇒ 70 digits.
4. log N = −3.4. What is the position of the first significant digit of N?
Characteristic −4 (since −3.4 = −4 + 0.6). Number of zeros right after the decimal = 4 − 1 = 3, so the first significant figure is in the 4th decimal place.

✏️ Practice — try these, take hints as needed

1. How many digits are in 3^40? (log 3 = 0.47712)
Compute 40 × log 3.
40 × 0.47712 = 19.0848.
digits = floor + 1.
20 digits
2. Number of digits in 2^100? (log 2 = 0.30103)
100 × log 2.
= 30.103.
Floor then +1.
31 digits
3. How many digits does 12^10 have? (log 2 = 0.30103, log 3 = 0.47712)
12 = 2^2 × 3, so log 12 = 2·log 2 + log 3.
log 12 = 1.07918; ×10 = 10.7918.
Floor 10, add 1.
11 digits
4. Digits in 7^15? (log 7 = 0.8451)
15 × log 7.
= 12.6765.
Floor 12, +1.
13 digits
5. log N = −2.1. How many zeros appear right after the decimal point before the first significant digit?
Characteristic is the next lower integer: −3.
Zeros = |characteristic| − 1.
3 − 1.
2 zeros

📝 Topic test — 8 questions

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