Formula

Number of digits from a common logarithm

Algebra • Logarithm • Characteristic and Mantissa

Formula

digits (N) = ⌊ lo g_{10} N ⌋ + 1

A positive integer

N

has

⌊ lo g_{10} N ⌋ + 1

digits, because

1 0^{k} \leq N < 1 0^{k + 1}

holds exactly when

N

has

k + 1

digits; the characteristic of

lo g_{10} N

is therefore one less than the digit count.

How to use it

1Compute $lo g_{10} N$ first, usually through the power rule, then take the integer part and add $1$ .
2Take the floor and never the nearest integer: a logarithm of $14.9$ still has characteristic $14$ .
3For a number smaller than $1$ the characteristic is negative and this digit rule does not apply.

Worked example

Given

lo g_{10} 2 = 0.3010

, how many digits has

2^{40}

Step-by-step

1
$lo g_{10} 2^{40} = 40 \times 0.3010 = 12.04$
Why: Bring the exponent down and multiply.
2
$⌊ 12.04 ⌋ + 1 = 13$
Why: The characteristic $12$ places the number between $1 0^{12}$ and $1 0^{13}$ .

Answer

13

digits

Watch out

The

+ 1

is what gets dropped, giving an answer one too small; rounding the logarithm up before flooring gives one too many. The mantissa fixes the leading digits, not the number of them.

Builds on

lo g a^{n} = n lo g a

Also known as

number of digits, characteristic and mantissa, floor log plus one, how many digits in a power

Number of digits from a common logarithm

Formula

Worked example

Appears in past papers

2020 (I)

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2019 (I)

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2016 (II)