Logarithms have a reputation for being confusing — but they're just a different way of writing exponents. Once you see that, everything clicks. Here's the whole picture, from the basic definition to the rules, graphs, and real examples.
A logarithm answers one simple question: what exponent do I need? If you know that 10² = 100, then log₁₀(100) = 2. It's the inverse of raising a number to a power. That's it — the entire idea fits in one sentence.
What log means
logₐ(x) = y exactly when aʸ = x. The log is the exponent you raise the base to, to get the number.
Product · Quotient · Power
Logs turn multiplication into addition, division into subtraction, and powers into multiplication.
Slow, steady growth
Log graphs rise quickly at first, then flatten out — never touching the y-axis.
What a Logarithm Actually Means
THE CORE IDEAA logarithm is the inverse of an exponent. If you know how to read 2³ = 8, you already know how to read log₂(8) = 3. They say the same thing from opposite directions.
Here's the formal definition, written both ways so you can see the connection:
Let's make that concrete with a few examples. Each row shows the logarithmic form and its exponential twin side by side:
| Logarithmic form | Exponential form | What it means in words |
|---|---|---|
| log₂(8) = 3 | 2³ = 8 | 2 raised to the power 3 gives 8 |
| log₁₀(1000) = 3 | 10³ = 1000 | 10 raised to the power 3 gives 1000 |
| log₅(25) = 2 | 5² = 25 | 5 raised to the power 2 gives 25 |
| log₃(1) = 0 | 3⁰ = 1 | Any base raised to 0 equals 1 — always |
| log₇(7) = 1 | 7¹ = 7 | Any base raised to 1 equals itself |
If you can rewrite a log as an exponent, you understand it. Try saying it out loud: "log base 2 of 8 is 3, because 2 to the 3rd power is 8." That verbal loop is the whole concept.
The Three Rules That Make Logs Useful
PRODUCT · QUOTIENT · POWERLogarithms aren't just a notation trick — they have three simple rules that turn hard multiplication and division into easy addition and subtraction. These are the rules you'll use in almost every problem.
1. The Product Rule
When you multiply two numbers inside a log, you can split it into the sum of two logs:
Example: log₂(4 · 8) = log₂(4) + log₂(8) = 2 + 3 = 5. Check: 4 · 8 = 32, and 2⁵ = 32. ✓
2. The Quotient Rule
When you divide inside a log, it becomes the difference of two logs:
Example: log₃(81 / 9) = log₃(81) − log₃(9) = 4 − 2 = 2. Check: 81 / 9 = 9, and 3² = 9. ✓
3. The Power Rule
When a number inside a log is raised to a power, that power moves out front as a multiplier:
Example: log₂(8³) = 3 · log₂(8) = 3 · 3 = 9. Check: 8³ = 512, and 2⁹ = 512. ✓
They're just exponent rules in disguise. Since aˣ · aʸ = aˣ⁺ʸ, the log of a product equals the sum of the logs. The quotient and power rules follow the same pattern from aˣ / aʸ = aˣ⁻ʸ and (aˣ)ⁿ = aˣⁿ.
How to Read a Logarithm Graph
SHAPE & BEHAVIOURA logarithm graph has a very distinctive shape: it climbs steeply at first, then flattens out and grows more and more slowly. It never touches the y-axis — that line is what mathematicians call a vertical asymptote.
Here are the key features to notice:
- Passes through (1, 0): logₐ(1) = 0 for any base, because a⁰ = 1.
- Passes through (a, 1): logₐ(a) = 1, because a¹ = a.
- Domain is x > 0 only: you can't take the log of zero or a negative number.
- Vertical asymptote at x = 0: as x gets closer to 0 from the right, the curve dives down toward negative infinity.
- Grows very slowly: to go from y = 1 to y = 2, x has to multiply by the base — that's a big jump for large bases.
Common Bases: Base 10 and Natural Log
Two bases show up more than any others, and each has its own notation and its own natural habitat.
| Name | Notation | Base | Where you'll see it |
|---|---|---|---|
| Common log | log(x) or log₁₀(x) | 10 | Science, engineering, pH, decibels, Richter scale |
| Natural log | ln(x) | e ≈ 2.718 | Calculus, physics, population growth, radioactive decay |
| Binary log | log₂(x) | 2 | Computer science, algorithm complexity, information theory |
When you see log(x) written with no base, it almost always means base 10. And ln(x) always means natural log — the base is Euler's number, e.
If you ever need to switch bases, this one formula handles it: logₐ(x) = log_b(x) / log_b(a). It works for any valid bases b. Most calculators only have log and ln buttons, so this is how you compute something like log₂(7).
Worked Examples
The best way to get comfortable with logs is to do a few by hand. Here are three that cover the main moves.
Example 1: Simplify a sum of logs
Simplify log₃(6) + log₃(4.5).
- Apply the product rule in reverse: log₃(6) + log₃(4.5) = log₃(6 · 4.5)
- Multiply inside: log₃(27)
- Recognise that 27 = 3³, so log₃(27) = 3
Answer: 3
Example 2: Solve an exponential equation
Solve 5ˣ = 200 for x.
- Take the log of both sides: log(5ˣ) = log(200)
- Apply the power rule: x · log(5) = log(200)
- Divide both sides: x = log(200) / log(5)
- Evaluate (using a calculator or change of base): x ≈ 3.29
Answer: x ≈ 3.29
Example 3: Use the power rule with a fractional exponent
Evaluate log₄(√16).
- Rewrite the square root as a power: log₄(16^(1/2))
- Apply the power rule: (1/2) · log₄(16)
- Since 16 = 4², log₄(16) = 2
- Multiply: (1/2) · 2 = 1
Answer: 1
Where Logarithms Show Up in Real Life
Logarithms aren't just a classroom exercise — they're hidden inside some of the most familiar measurement scales in daily life.
- The Richter scale measures earthquake magnitude on a logarithmic scale — each whole number step is ten times more powerful than the last.
- The pH scale for acidity is logarithmic, which is why a pH of 4 is ten times more acidic than a pH of 5.
- Decibels for sound intensity work the same way — a 10 dB increase sounds about twice as loud, but represents ten times the energy.
- Population growth and radioactive decay follow exponential curves, so solving for time requires logarithms.
- Computer science uses log₂ to describe how many steps a binary search or balanced tree takes as data grows.