Grade 11 / High 2 (age 16-17)

high school Common Logarithm

A common logarithm has base 10. The common log of a positive number N splits into an integer part, the characteristic, and a fractional part that is at least 0 and less than 1, the mantissa. Because log N goes up by exactly 1 every time N is multiplied by 10, the characteristic tracks how many digits N has, and the mantissa carries the order of those digits. That is why log 2 alone tells you how many digits 2³⁰ has, without multiplying it out. Move x in the picture and the bar on the vertical axis splits into a white part, the characteristic, and a gold part, the mantissa.

Logarithms with base 10

Number x200

👀 What to read off the graph

①A common log leaves out the base 10 and is written log x

②The marks 1, 10, 100, 1000 are equally spaced, and the log values there are 0, 1, 2, 3

③Each time x is multiplied by 10, the log value rises by exactly 1

④So for 2 ≤ x < 1000 the log splits cleanly into an integer characteristic and a mantissa between 0 and 1

Characteristic and mantissa

Splitting a log in two

log N = (characteristic) + (mantissa), 0 ≤ mantissa < 1

The characteristic is an integer; the mantissa is the fractional part, at least 0 and less than 1

Digits and the characteristic

integer N has d digits ⇔ characteristic = d − 1 ⇔ 10^d-1 ≤ N < 10^d

If a number N ≥ 1 has d digits, its characteristic is d − 1

Same mantissa, same digits

🔢 Shifting by powers of 10

①log 200 = log 2 + 2 and log 0.02 = log 2 − 2: multiplying or dividing by a power of 10 changes only the characteristic

②So 2, 20, 200 and 2000 all have the mantissa 0.3010

③The characteristic decides where the decimal point goes, and the mantissa decides the order of the digits

Worked examples

Example 1

Given log 2 = 0.3010, find log 200.

Write 200 as 2 × 10².

log 200 = log 2 + log 10²

The log of a product is the sum of the logs, and log 10² = 2.

= 0.3010 + 2 = 2.3010

▸ 2.3010 (characteristic 2, mantissa 0.3010)

Multiplying by 10² leaves the mantissa 0.3010 alone and moves the characteristic from 0 to 2.

Example 2

How many digits does 2³⁰ have? (log 2 = 0.3010)

Taking the common log of 2³⁰ brings the exponent 30 to the front.

log 2³⁰ = 30 log 2 = 30 × 0.3010 = 9.03

The log value 9.03 has characteristic 9, so 10⁹ ≤ 2³⁰ < 10¹⁰.

characteristic 9 ⇒ 10 digits

▸ 10 digits

Indeed 2³⁰ = 1,073,741,824, which has 10 digits. A characteristic of 9 means the number is at least 10⁹ and less than 10¹⁰.

Summary

Key ideas

log N = characteristic + mantissa, d digits ⇔ characteristic = d − 1

The characteristic handles how many digits; the mantissa handles which digits

2021 CSAT Math type, adapted

How many digits does 3²⁰ have? (log 3 = 0.4771)

①9 digits

②10 digits

③11 digits

④12 digits

⑤13 digits

▸ ② 10 digits

Take the common log of 3²⁰ and read off its characteristic.

log 3²⁰ = 20 × 0.4771 = 9.542

The characteristic is 9, so 10⁹ ≤ 3²⁰ < 10¹⁰, which means 10 digits.

characteristic 9 ⇒ 10 digits

🎯 Exam Points

①log N = characteristic + mantissa (0 ≤ mantissa < 1)

②N ≥ 1 has d digits ⇔ characteristic = d − 1

③Digits of a power = characteristic of its log + 1

④Two numbers with the same mantissa have the same digits and differ by a power-of-10 factor

⑤If 0 < N < 1 the characteristic is negative, and a characteristic of −k means the first nonzero digit is in the kth decimal place

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Last updated 2026-10-17

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