Natural Log Studio
Calculate logarithms, visualize exponential growth/decay, and explore standard values.
Logarithm Solver
Calculate the logarithm of any number. Use base e for Natural Log (ln).
The Magic of Natural Logarithms
Most people are familiar with "Log base 10" because we have 10 fingers. But the universe doesn't count on its fingers; it grows continuously.Natural Log (ln) is the language of that growth.
Natural Growth
Whether it's bacteria dividing in a petri dish or your money compounding in a bank, things grow proportional to their current size. **e** describes this perfect, continuous growth.
Solving for Time
If you know how fast something is growing (rate) and how much you have (amount), how long did it take? **ln** helps you work backwards to find **Time**.
Decay
It works in reverse too! Radioactive dating and cooling coffee follow the same laws, just with a negative rate. ln allows us to predict when they will reach a stable state.
Ln vs Log: The Showdown
New students often confuse log and ln. Here is the Cheat Sheet:
- log(x): Default is Base 10. Used for pH levels, Richter scale (earthquakes), and Decibels (sound). It shrinks numbers very aggressively (log 1,000,000 is just 6).
- ln(x): Default is Base e (~2.718). Used for calculus, biology, physics, and finance. It is the "natural" way changes happen in the continuity of time.
Try It Yourself
Go to the Log Calculator tab above. Enter "10" as the Base and "100" as the Number. The result is 2 (because 10² = 100). Now try standard ln (base e) for 7.389... the result is approx 2 (because e² ≈ 7.389).
Frequently Asked Questions
What is the difference between log and ln?
log usually refers to base 10 (common logarithm), used in engineering and sound scales (decibels). ln refers to base e (approx 2.718), used in calculus, population growth, and physics. They are proportional: ln(x) ≈ 2.303 * log(x).
What is 'e' exactly?
e (Euler's Number) is approx 2.71828. It is the universal limit of growth. If you take $1 and double it continuously (100% interest compounded infinitely fast), you get $2.718...
Why is ln(1) = 0?
Because any number raised to the power of 0 is 1. Since ln(x) asks 'e to what power equals x?', and e^0 = 1, the answer is 0.
Why is ln of a negative number undefined?
You cannot raise a positive number (like e) to any power and get a negative result. Try it: e^2 is positive, e^-100 is tiny but still positive. The graph never touches numbers ≤ 0.
How do I convert between e^x and ln(x)?
They are inverses. If y = e^x, then x = ln(y). This effectively 'undoes' the exponent operation, helping you solve for time in growth equations.
What is the Rule of 72?
It's a shortcut to find doubling time. Time = 72 / Interest Rate. It works because ln(2) ≈ 0.693, closely related to 72%.
How is Natural Log used in finance?
It is used for Continuous Compound Interest. The formula A = Pe^(rt) uses e and ln to calculate returns over time with instantaneous compounding.
What is the derivative of ln(x)?
The derivative is simply 1/x. This unique property makes ln(x) incredibly important in integration and solving differential equations.
Can ln(x) be bigger than x?
No. For all positive real numbers, ln(x) is always smaller than x. The curve captures the idea of 'diminishing returns'.
Who discovered logarithms?
John Napier introduced them in 1614 to simplify complex multiplication calculations into additions.