CDS Mathematics · Logarithms

Counting Digits with Logarithms

A number whose common logarithm has integer part k has k + 1 digits before the decimal point.

All Logarithms notes CDS Mathematics strategy

Why this matters

Ten PYQs, one of them HARD. Take the log of the power, keep the integer part, add one. The mirror-image question — how many zeros after the decimal point in (0.5)¹⁰⁰⁰ — uses the same log and reads it the other way.

Concept 1 of 1: Digits and leading zeros

Every number from

10^k

up to just below

10^{k+1}

has

k + 1

digits, and its log lies between

k

and

k + 1

. So the integer part of the log counts the digits, less one.

Definition

$N \ge 1$ : digits $= \lfloor\log_{10}N\rfloor + 1$ .
$N < 1$ : if $\log_{10}N = -m.f$ (so $N = 10^{1 - 0.f} \times 10^{-(m+1)}$ ), the first significant digit is in place $m + 1$ , after $m$ zeros.
Write the base in primes first: $108 = 2^2 \cdot 3^3$ , $125^{100} = 5^{300} = \dfrac{10^{300}}{2^{300}}$ .
A number between $100$ and $1000$ has log between $2$ and $3$ ; a positive number below $1$ has a negative log.

Number of digits

\lfloor \log_{10} N \rfloor + 1

Worked example

How many digits has

2^{50}

? (

\log 2 = 0.301

)

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (I) 2017 — Elementary Mathematics · Q6Easy

Example 1 · Logarithms · Number of Digits and Characteristic

What is the number of digits in

2^{40}

? (Given that

\log_{10} 2 = 0.301

)

Add one

\log_{10}2^{40} = 12.04

, so

2^{40}

has

13

digits, not

12

. The characteristic is one less than the digit count.

Zeros after the point: use the next power down

\log N = -17.47

means

N = 10^{0.53} \times 10^{-18}

: the first significant digit is in the 18th place, after

17

zeros. Reading

-17.47

as '18 zeros' is the slip.

Drill 9 more on digits and leading zeros

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Digits and leading zeros
Number of digits
$\lfloor \log_{10} N \rfloor + 1$

Watch out for (2)

Add one
→ Digits and leading zeros
Zeros after the point: use the next power down
→ Digits and leading zeros

Test yourself on Logarithms

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Logarithms chapter test Drill all 10 questions