The Natural Logarithm
“A logarithm is a backwards exponent counting 'how many times multiplied'.”
The formula
ln x = y ⇔ eʸ = xHow to read it: ln x answers 'to what power must e be raised to get x'
- ln x
- — the logarithm with base e (the natural log)
- e
- — the constant e ≈ 2.718
- y
- — the exponent for which e raised to y gives x
The hook
Logs aren't hard — they're just backwards exponents asking 'how many times must I multiply to reach this'.
In plain words
ln x answers 'to what power must e be raised to get x'. In other words, it's the inverse of the exponential eˣ.
The intuition
An exponent asks 'repeat multiplication how many times to reach this size?'; a log asks the reverse, 'how many times to become this size?'. That's why logs turn multiplication into addition — like counting digits.
How it's built
It's the special log with base e. Saying ln x = y is the same as saying eʸ = x. Exponent and log are just the same fact read from two sides.
Example
ln 1 = 0 (since e⁰=1), ln e = 1 (e¹=e), ln(e²) = 2. Just count 'how many times e was multiplied'.
Common mistake
ln and log are easy to mix up. A bare log usually means base 10 (the common log), while ln has base e.
Where it's used
Solving for 'how long it takes (time)' in growth and decay, simplifying calculations by turning products into sums, and appearing in calculus as the integral of 1/x.
Where it came from
Napier invented logarithms to turn multiplication into addition, revolutionizing astronomical computation; the base-e 'natural' log settled in as the smoothest form for calculus.
Prerequisites
Quick check
What is the value of ln e?
- 0
- 1✓
- e
- 2.718
Practice
What is ln 1?
Answer: 0
- 'e to what power is 1?'
- e⁰=1, so ln 1 = 0
Takeaway: Anything to the 0 power is 1 — so ln 1 = 0.
What is ln(e³)?
Answer: 3
- 'e to what power is e³?'
- It's e³, so ln(e³) = 3
Takeaway: ln(eⁿ)=n — it pulls the exponent straight out.
Evaluate ln(e·e²) using log properties.
Answer: undefined
- ln(ab)=ln a+ln b → ln e + ln e² = 1 + 2 = 3
- Check: e·e²=e³, so ln(e³)=3
Takeaway: Logs turn products into sums — that's their power.