Logarithms

A logarithm is the exponent — nothing more, and nothing less.

Keep doubling: 2, 4, 8, 16, 32, 64...

How many doublings does it take to pass a thousand? Nine gives 512, ten gives 1024.

2 to what power reaches a thousand? 2 1 4 2 8 3 16 4 32 5 64 6 128 7 256 8 512 9 1024 10 1000 the exponent is what we are hunting for between 9 and 10 — a log is that exponent

So the answer sits between 9 and 10. We know what the question is; we just have no symbol for the answer.

Every equation so far has hunted for the base or the result. This one hunts for the exponent.

That answer is written log2 1000. A logarithm is not a new operation — it is the name of the exponent you were already looking for.

The same fact, written two ways exponential form 2⁵ = 32 logarithmic form log₂ 32 = 5 base 2, exponent 5 base 2, answer 5 a logarithm asks for the exponent

The two forms are one fact

25 = 32 and log2 32 = 5 say exactly the same thing, arranged differently.

Reading it aloud helps: "log base two of thirty-two is five" means "two to the fifth is thirty-two".

Every log problem can be turned into an exponent problem, and that is usually the way out.

The base stays the base

In both forms the little number is the base. The log is always the exponent, never the base and never the answer inside.

The values you should know cold

logb 1 = 0 for every valid real base b>0, b ≠ 1, because b0=1.

logb b = 1, because one copy of the base gives the base. And logb bk = k.

These are not rules to memorise separately — each is just the definition read back.

Estimate before you calculate

log2 100 must lie between 6 and 7, since 26 = 64 and 27 = 128. A quick bracket catches a calculator slip instantly.

The three laws for positive arguments, and where they come from

log(MN) = log M + log N. Multiplying the numbers adds the exponents, which is the exponent rule wearing a different hat.

logMN = log M - log N, for the same reason with division.

log Mp = p log M, because a power is repeated multiplication and so repeats the addition.

There is no law for a sum

log(M + N) does not split up. Try log2(4+4) = 3 against log2 4 + log2 4 = 4 — they disagree.

Solving equations with the unknown upstairs

When x is an exponent, take the log of both sides. The power law then brings it down to ground level.

3x = 20 becomes x log 3 = log 20, so x = log 20log 3 ≈ 2.73.

Any valid real base (b>0, b ≠ 1) works as long as you use the same one on both sides; base 10 or e are on the calculator.

Why this is the whole trick

The power law is the only tool that moves an exponent. Every "solve for the exponent" problem uses it.

Logs of negatives do not exist

logb x asks what power of a positive base gives x. A positive base raised to any power stays positive.

So the argument of a log must be strictly greater than zero — never zero, never negative.

This matters when solving: an answer that makes any argument negative has to be discarded.

Check the argument, not just the answer

Solving log(x-3) = 1 gives x = 13, which is fine. But a solution of x = 1 would make the inside -2, and must be thrown out.

Worked examples

Reading a log

Evaluate log3 81.

  1. The question is: three to what power gives 81? A log is the exponent.
  2. 34 = 81. Count up: 3, 9, 27, 81.
  3. So log3 81 = 4. No calculator needed once the question is read properly.

4

Condensing

Write 2log x + log 3 - log y as a single logarithm.

  1. The 2 becomes a power: 2 log x = log x2. The power law, used backwards.
  2. Addition becomes multiplication: log x2 + log 3 = log 3x2. The product law.
  3. Subtraction becomes division: log 3x2y. The quotient law.

log 3x2y

Unknown in the exponent

Solve 5x = 90, to two decimal places.

  1. Take log of both sides: log 5x = log 90. The same operation on both sides keeps the equation true.
  2. Bring the power down: x log 5 = log 90. The power law is what frees the exponent.
  3. So x = log 90log 5 ≈ 2.80. Check: 52 = 25 and 53 = 125, so a value near 2.8 is right.

x ≈ 2.80

Words to know

Logarithm
logb y is the exponent that turns b into y.
Common log
Base 10, usually written log with no base shown.
Natural log
Base e ≈ 2.718, written ln.
Change of base
logb x = log xlog b, which is how a calculator handles any base.
Argument
The thing inside the log. It must be strictly positive.

Mistakes to watch for

Splitting the log of a sum. log(M+N) is not log M + log N. Test it: log2(4+4) = 3 but log2 4 + log2 4 = 4. Only products, quotients and powers have laws.

Reading log 20log 3 as log203. Division of two logs is not the log of a quotient — the first is about 2.73, the second about 0.82.

Taking the log of a negative or zero. log(x-3) needs x > 3, so any solution at or below 3 must be discarded however good the algebra looked.

Confusing the base with the answer. In log2 32 = 5, the 2 is the base and the 5 is the exponent. Writing log32 2 asks an entirely different question.

Forgetting to apply the power law before solving. 3x = 20 does not give x = 203. Taking logs and bringing the exponent down is the only route.

Napier published logarithms in 1614 to make multiplication easier. Turning multiplication into addition cut the labour of astronomy so sharply that it was said to double an astronomer’s life.

The slide rule, which put people on the moon, is a logarithmic scale. Sliding two of them adds the logs, which multiplies the numbers.

Earthquakes, sound and acidity are all measured on log scales, because each is easier to talk about in terms of "how many times bigger".

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