Free maths tool · Level: Ages 16 to 18
Logarithms
Slide x along the curve. Its height is always the exponent you have to put on the base to land back on x.
A logarithm is an exponent. log(1000) = 3 says nothing more than 10³ = 1000, so asking for a log is asking "to what power". Written without a base, log means base 10, and ln means base e. The logarithm of 0 or of a negative number does not exist, because no power of a positive base ever lands there.
Read the graph as an exponent
The tool puts one point on the curve and writes two lines beside it that say exactly the same thing: log(x) = y, and the base to the power y is x. While those two lines feel like different facts, the topic stays hard; once they collapse into one, most of the work is done.
Set the base to 2 and slide to 8. The height is 3, because 2³ = 8. Switch to base 10 and the same x gives about 0.9, because even a small power of 10 is already large. That is also why the curve climbs more slowly as the base grows.
The rules are differences in height
Switch on the layer with two points and watch the vertical distance. It belongs to a doubling of x, and it does not change as you slide x along. That is precisely log(2x) = log(x) + log(2).
The same reasoning gives the others. Dividing turns into subtracting, and a power moves to the front: log(x³) is three times log(x), because you are stacking the same height three times. Seen that way, there is nothing left to memorise.
Why there is nothing left of the axis
Switch the asymptote on and the vertical line x = 0 appears, with the curve never touching it. However close to zero you get, the exponent simply dives further, without ever reaching a final value.
To the left there is nothing at all, and that is not a shortcoming of the drawing. A positive base raised to any power stays positive, so no exponent can produce zero or a negative number. Ask a calculator for log(-5) and you get an error rather than a number.
Do logarithms still feel like a list of rules?
A tutor shows where each rule comes from, working from your own exercises.
Frequently asked questions
- What is a logarithm in plain words?
- The answer to "what power do I raise the base to". log(100) = 2 because 10² = 100. The logarithm is itself an exponent, not some new kind of number.
- How do you calculate a logarithm?
- If the number sits neatly on a power of the base you can read it off: log₂(32) = 5 because 2⁵ = 32. Otherwise use the log or ln key on a calculator. For any other base, use the change of base rule: log₃(7) is log(7) divided by log(3).
- What is the difference between log and ln?
- Only the base. Written on its own, log means base 10, and ln is the natural logarithm with base e, about 2.72. Every rule is identical for both.
- What are the logarithm rules?
- A product becomes a sum, a quotient becomes a difference, and a power moves to the front: log(a·b) = log(a) + log(b), log(a/b) = log(a) - log(b) and log(aⁿ) = n·log(a). All three hold only for strictly positive a and b.
- Why does the logarithm of a negative number not exist?
- Because no exponent manages it. A positive base stays positive whatever power you use, so you never reach zero or a negative number. The domain of the logarithm is therefore everything strictly greater than zero.
More interactive maths tools
- Derivative and tangent →Drag the point and watch the tangent tilt, with the secant closing in on it.
- Discriminant and parabola →Slide a, b and c and see D decide whether the parabola cuts, touches or misses the axis.
- Integral and Riemann sum →Push the rectangle count up and watch the sum close in on the exact area.