SumNinja

Find the missing power with logarithms

About 4 min

A logarithm finds a missing power. Four rounds of doubling turn one point into sixteen, so log₂(16) = 4.

A game score starts at one and doubles after each round. You know it reached sixteen. To find the number of rounds, work backwards from that score. A logarithm does that job. The small 2 in log₂ tells you to use powers of two.

Recover the number of doubling rounds

log₂(16) = ? → 2^? = 16

  1. Translate the notation

    log⁡2(16)=?⟺2?=16\log_2(16)=?\quad\Longleftrightarrow\quad2^{?}=16

    The base is the factor you use again and again. With base two, ask which power of two makes sixteen.

  2. Count the doubling steps

    1→2→4→8→161\to2\to4\to8\to16

    There are four changes, starting from one. The starting value is not a doubling round.

  3. Check by taking the power

    24=16⇒log⁡2(16)=42^4=16\quad\Rightarrow\quad\log_2(16)=4

    Four factors of two rebuild the score. A logarithm returns that exponent, which is the missing power.

The target score goes across; rounds go up

One point needs zero rounds, two need one, four need two and eight need three. Doubling the target adds one to the height.

Make a graph from the same question

For y = log₂(x), let x be the target score and y the rounds needed. A score of eight needs three rounds, so (8, 3) is a point on the graph.

Scores 1, 2, 4 and 8 give rounds 0, 1, 2 and 3. Each doubling moves the height up by one. The gaps between scores grow, so the curve rises more slowly.

A power of two stays above zero. It cannot give zero or a negative target, so this graph needs x above zero. Half a point means going one doubling step backwards: 2⁻¹ = 1/2, so log₂(1/2) = −1.

Change the base, not the meaning

With base three, each step triples the start. Since 3² = 9, log₃(9) = 2. The job stays the same: recover the power. You can also place a target between known powers without finding its exact value.

Try it yourself

Work out an answer and explain your reasoning. Then check the explanation.

Change the target

log⁡2(32)\log_2(32)

How many doubling rounds reach this target from one?

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55

2⁵ = 32, so the missing power is 5.

Read a different base

log⁡3(81)\log_3(81)

What power of three reaches the target? Check by multiplying back.

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44

3 × 3 × 3 × 3 = 81.

Estimate between anchor powers

log⁡2(12)\log_2(12)

Use the powers just below and above twelve to place the missing power.

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3<log⁡2(12)<43<\log_2(12)<4

Since 2³ = 8 and 2⁴ = 16, the power that makes twelve lies between three and four.

Choose the rounds-from-score graph

y=log⁡2(x)y=\log_2(x)

Which graph gives the right number of rounds when the target score is one?

Graph A

At scores one, two and four, the heights are zero, one and two.

Graph B

At scores one, two and four, the heights are two, three and four.

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Graph A\text{Graph A}

A score of one needs zero doubling rounds. Graph A includes (1, 0). Graph B has an extra two in every height.

Why a logarithmic scale can help

An ordinary scale uses equal gaps for equal additions: one, two, three, four. A base-two log scale uses equal gaps for doubling: one, two, four, eight. This lets a small space show very large changes.

A log scale changes an axis's spacing. The graph above instead plots a logarithm on ordinary axes. The ideas connect, but they are not the same picture.

Read a shifted or flipped log graph

A number in front of the log multiplies its result. In y = 2log₂(x), x = 4 gives 2 × 2 = 4.

In y = log₂(x) + 2, add two to every height. So x = 1 gives y = 2. A minus sign before the log turns the curve upside down. Check a known power in the full rule before choosing.

Try the graph connection

The game asks you to match logarithm rules to graphs. Use a target that is a power you know, then check its height. Include any changes in the full rule.

2-minute practice · keep going after mistakes · opens in a new tab

Try logarithmic graph questions