Before the method: the idea behind it
The question an exponent cannot answer
You can already work out what 2⁵ is. But if a population doubles every year and you want to know when it will be ten times bigger, you need the reverse: not “what is 2 to the power 5”, but “2 to what power gives 10?”. That question has an answer, it has a name, and once you can compute it a whole class of previously unsolvable problems opens up.
Part one
A logarithm is an exponent
That sentence is the whole definition. Everything else follows from it.
When you write \(\log_2 32 = 5\), you are saying “the power you raise 2 to in order to get 32 is 5”. The logarithm is the exponent. So every logarithm statement is an index statement wearing different clothes, and being able to switch between the two forms instantly is the single most useful skill in this topic.
Read the arrow both ways. Given \(3^4=81\) you can immediately write \(\log_3 81=4\). Given \(\log_5 125=3\) you can immediately write \(5^3=125\). When a logarithm question looks impenetrable, converting it to index form is almost always the first move.
Two consequences are worth memorising because they appear constantly: \(\log_a 1=0\), since anything to the power 0 is 1, and \(\log_a a=1\), since anything to the power 1 is itself.
One restriction matters. You can only take the logarithm of a positive number, because a positive base raised to any power is always positive. There is no exponent that turns 2 into −8, so \(\log_2(-8)\) does not exist. This is why solutions must be checked at the end of every logarithmic equation.
Part two
Inverses, and what that looks like
If exponentials and logarithms undo each other, their graphs must be reflections in the line y = x.
Every point (a, b) on the exponential curve corresponds to a point (b, a) on the logarithmic one, with the coordinates simply swapped. Drag the point below and watch its partner move.
The coordinates swap. Notice too that the exponential’s horizontal asymptote at y = 0 becomes the logarithm’s vertical asymptote at x = 0.
The reflection explains the shape of every log graph you will draw. The exponential is defined for all x but only ever outputs positive values; so the logarithm accepts only positive inputs but outputs any value. The exponential flattens towards y = 0 on the left; the logarithm plunges towards negative infinity as x approaches 0 from the right. And \(y=\log_a x\) always passes through (1, 0), because \(a^0=1\).
Growth is deceptively slow. It takes x = 1024 for \(\log_2 x\) to reach 10, and over a million to reach 20. That sluggishness is exactly what makes logarithms useful for taming enormous numbers.
These three laws are index laws in disguise. Multiplying powers means adding exponents, so multiplying inside a logarithm means adding logarithms. That correspondence, turning multiplication into addition, is why logarithms were invented in the first place, and it is what the third law exploits when you need to bring an unknown down from an exponent.
Part three
Measuring things that span orders of magnitude
Some quantities in nature vary by factors of billions. A logarithmic scale makes them manageable.
The quietest audible sound and a jet engine differ in intensity by a factor of about a trillion. No linear scale can display both usefully. But take the logarithm and that trillion-fold range becomes the numbers 0 to 120, which is exactly the decibel scale.
Sound intensity compared with the quietest audible sound. Switch between scales:
On a linear scale the quiet sounds are invisible: the jet engine’s bar is a trillion times longer than the whisper’s.
The same trick appears wherever ranges are vast. The Richter scale for earthquakes, pH for acidity, and stellar magnitudes for brightness are all logarithmic. Each step on these scales means multiplying by a fixed factor, so a magnitude 7 earthquake releases about 100 times the energy of a magnitude 5, and a pH of 3 is 100 times more acidic than a pH of 5.
This is why “the earthquake was twice as strong” is almost always wrong when quoted from Richter numbers: the scale does not add, it multiplies.
Where this shows up in real life
Carbon dating and half-lives
To find the age of a sample you must solve for the exponent in a decay equation, which requires a logarithm. Every radiocarbon date, every drug clearance calculation, and every compound interest question asking “how long?” is solved this way.
Why the numbers on your dial are logarithmic
Human perception itself is roughly logarithmic: a sound must be about ten times more intense to seem twice as loud. Volume controls, camera exposure stops and musical octaves are all built on this, matching equal steps on the dial to equal multiplications in the physical quantity.
Before you start
What to reach for
| The question involves… | Use |
|---|---|
| evaluating a logarithm such as log₃81 | Convert to index form: 3 to what power is 81? |
| combining or splitting logarithms | The three log laws |
| an unknown in an exponent | Take logs of both sides, then use log(mᵏ) = k log m |
| an unknown inside a logarithm | Combine into one log, convert to index form, then check for invalid solutions |
| a logarithm with an awkward base | Change of base, or the CAS |
| sketching y = log(x − c) + b | Asymptote at x = c, then find the x-intercept |
Next: the working method, twenty practice questions, ten harder ones, and the ClassPad guide.