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Digits and logs

How many digits has 2¹⁰⁰? What does 7⁴⁰ start with? Work it out from log 2, log 3 and log 7 — no calculator, and the answer is exact.

  • Number sense
  • Untimed
  • Endless levels

How it plays

A power or a product of powers; say how many digits it has, what its first digit is, or its log to 2 decimal places. Three anchors do all of it: log 2 ≈ 0.3010, log 3 ≈ 0.4771, log 7 ≈ 0.8451, and log 5 = 1 − log 2.

A number with log 30.1 lies between 10³⁰ and 10³¹, so it has 31 digits — the whole part plus one. The fractional part says where it sits between: 10^0.1 is about 1.26, so it starts with 1.

The digit counts and first digits are exact, computed from the number written out in full; the exponents are chosen so the 4-dp anchors give the true answer. Logs are asked to 2 dp and accepted within 0.01. The wrong options are the real slips: the +1 forgotten, log 2 taken as 0.3, the first decimal read as the first digit.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This makes estimating the size and first digit of powers from a few remembered logs quicker and surer. It claims nothing wider.

Learn

How to get better at digits and logs: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Write the number as primes to powers: 6²⁰ is 2²⁰ × 3²⁰, 12ⁿ is 2²ⁿ × 3ⁿ, 6,000 is 2 × 3 × 10³.
  2. Take its log: each power multiplies its prime’s log, the logs add, and a power of ten adds a whole number. Use all four decimal places of the anchors.
  3. Split the result into its whole part and its fractional part. The whole part says how big; the fractional part says which digits.
  4. Answer the question asked: digits are the whole part + 1; the first digit comes from the fractional part; a log is the whole thing, to 2 dp.

Techniques and tricks

  • Digits are the whole part plus one

    A number with log 30.1 lies between 10³⁰ and 10³¹. 10³⁰ already has 31 digits, so the count is the whole part of the log plus one.

    e.g. 2¹⁰⁰: 100 × 0.3010 = 30.10, so 31 digits

  • Multiply the anchor in two parts

    n × 0.3010 is n × 0.3 plus n × 0.001. The small part is what the crude 0.3 throws away, and it decides close cases.

    e.g. 2⁶⁴: 64 × 0.3 = 19.2, plus 0.064, makes 19.264, so 20 digits

  • The leading-digit ladder

    Learn where each first digit begins: log 2 = 0.301, log 3 = 0.477, log 4 = 0.602, log 5 = 0.699, log 6 = 0.778, log 7 = 0.845, log 8 = 0.903, log 9 = 0.954. The fractional part falls on one rung.

    e.g. 7⁴⁰: 40 × 0.8451 = 33.804. 0.804 lies between 0.778 (6) and 0.845 (7), so it starts with 6.

  • Build the other logs from three

    Every log you need comes from log 2, log 3 and log 7: log 4 = 2 log 2, log 5 = 1 − log 2, log 6 = log 2 + log 3, log 8 = 3 log 2, log 9 = 2 log 3.

    e.g. 5³⁰: log 5 = 1 − 0.3010 = 0.6990; 30 × 0.6990 = 20.97, so 21 digits

  • Split off the power of ten

    For the log of a round number, write it as a small number times 10ᵏ. The small number gives the decimals; k is added to the whole part.

    e.g. log 6,000 = log 6 + 3 = 0.3010 + 0.4771 + 3 = 3.7781, so 3.78

  • 2¹⁰ ≈ 10³ as a check

    2¹⁰ = 1,024, a little over a thousand. So 2ⁿ has a little more than 0.3n + 1 digits, and that is a quick check on your multiplication.

    e.g. 2¹⁰⁰ = (1,024)¹⁰ ≈ 1.27 × 10³⁰: 31 digits, starting with 1

As the levels climb

  • You start with the digits of powers of 2. Learn 0.3010 cold and practise multiplying it in two parts.
  • Then powers of 3 and 7, with 0.4771 and 0.8451; then products like 6ⁿ, 12ⁿ, 14ⁿ and 5ⁿ, where you add several logs first. Write the primes out before you multiply anything.
  • Then the leading digit, where only the fractional part matters. The exponents are chosen so the fractional part sits clear of every rung of the ladder, so four-decimal anchors always land on the true digit.
  • Last, logs to 2 dp of numbers like 6,000 or 63,000; a typed answer within 0.01 counts. Factor into 2s, 3s, 7s and a power of ten, and add.
  • Between the new kinds, the exponents keep growing — from tens into the hundreds. A slip in the third decimal place of your product can now move the whole part, so the 0.001 parts matter more; you get a little more time each step.

Traps to avoid

  • The +1 forgotten. A log of 30.10 is 31 digits, not 30; the count one short is the first slip the options are built from.
  • Crude anchors. Log 3 taken as 0.5 makes 3⁵⁰ “25.0, so 26 digits”; with 0.4771 it is 23.86, so 24. Log 2 as 0.3 drifts more slowly, but it drifts.
  • The first decimal read as the first digit. 7⁴⁰ has log 33.804, and “8” is offered; the digit comes from where 0.804 falls on the ladder, which is 6.
  • A log with something missing: the power of ten left off (0.78 for log 6,000), each prime counted once instead of times its power (log 9 as log 3), or the digit count given for the log.
  • A natural log offered for log₁₀. The game asks for base 10; a natural log is about 2.3 times bigger.

How to practise

  • Keep “Anchor logs” on “Shown in every question” until the method is fluent, then switch to “From memory” — four numbers are worth carrying: 0.3010, 0.4771, 0.6990 and 0.8451.
  • Use “Kinds of question” to drill “The leading digit” on its own once digit counts are easy; it is the one that needs the ladder.
  • Switch “Answer by” to “Typing the answer” so the near-miss options cannot do the checking for you. If you keep choosing, raise “Options shown” to 6 so the +1 slip, the crude anchor and the missing power of ten all appear together.

Levels

It is untimed unless you turn the clock on. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More number sense

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