Natural Log Calculator
Compute the natural logarithm of any positive number, reverse-solve to find x given ln(x), convert between log bases, or apply ln to doubling time and half-life problems. Enter a value and get instant results with a worked step-by-step breakdown.
Formula
Worked example
For x = 10: ln(10) asks what power of e gives 10. Since e^2.302585 ≈ 10, ln(10) ≈ 2.302585. Reversing: e^2.302585 = 10. Base change: log₂(10) = ln(10)/ln(2) = 2.302585/0.693147 ≈ 3.321928. Doubling time at 7%: ln(2)/0.07 = 9.9 years.
What the natural logarithm means
The natural logarithm of a positive number x is the exponent to which the constant e, Euler's number (approximately 2.71828), must be raised to produce x. Written ln(x) or log base e, it is the inverse of the exponential function e^x. Because e arises naturally in continuous growth, compound interest, radioactive decay, and countless physical processes, the natural log is the default logarithm throughout calculus, statistics, and the sciences. Three anchor values are worth memorizing: ln(1) = 0, ln(e) = 1, and ln(2) = 0.693147.
Key properties you can apply immediately
Natural logs turn multiplication into addition: ln(ab) = ln(a) + ln(b), and ln(a/b) = ln(a) - ln(b). The power rule states ln(x^n) = n * ln(x), which is why logs make exponent equations tractable. The base-change formula log_b(x) = ln(x) / ln(b) lets you convert any logarithm into natural logs, or vice versa. The inverse property e^(ln x) = x is used to solve for unknowns in exponential models: isolate the exponent, take the natural log of both sides, and the answer falls out.
Why it is undefined for zero and negatives
The exponential function e^y is always strictly positive; no real exponent makes e equal zero or a negative number. Because ln reverses that function, it only accepts positive inputs. As x approaches 0 from above, ln(x) plunges toward negative infinity; for x at or below zero the result is either undefined or complex. This calculator requires x greater than 0 in forward mode and returns no result otherwise. For complex logarithms (ln of a negative number) the answer involves an imaginary component not covered here.
Doubling time and half-life: ln(2) in action
Two of the most practical applications of the natural log are doubling time and half-life. For a quantity growing continuously at rate r (as a decimal), the time to double is T = ln(2)/r = 0.693147/r. The popular Rule of 70 is simply this formula approximated: 70/r% gives the answer in years when r is small. For a decaying quantity losing a fraction d per year, the half-life is T = ln(2)/d by the same formula. These results apply directly to compound interest, population dynamics, radioactive isotopes, drug clearance from the body, and capacitor discharge in electronics.
Logarithms in different bases
While the natural log (base e) is dominant in calculus and continuous mathematics, two other bases appear constantly. The common logarithm (base 10) is used in pH, decibels, earthquake magnitude (Richter scale), and many engineering contexts. The binary logarithm (base 2) drives information theory, computer science (bits and binary search depth), and signal processing. The base-change formula log_b(x) = ln(x) / ln(b) converts between any two bases, and this calculator applies it directly when you select Log base b mode.
Natural log reference values and properties
| x or expression | ln(x) or result | Notes |
|---|---|---|
| x approaching 0+ | approaches -inf | No lower bound |
| 0.1 | -2.302585 | Equal to -ln(10) |
| 0.5 | -0.693147 | Equal to -ln(2) |
| 1 | 0 | ln(1) = 0 always |
| 2 | 0.693147 | Basis of doubling-time formula |
| e ≈ 2.71828 | 1 | ln(e) = 1 always |
| 10 | 2.302585 | ln(10); used in base-10 conversion |
| 100 | 4.605170 | 2 * ln(10) |
| e^10 | 10 | Inverse: ln(e^k) = k |
| ln(ab) | ln(a) + ln(b) | Product rule |
| ln(a/b) | ln(a) - ln(b) | Quotient rule |
| ln(x^n) | n * ln(x) | Power rule |
| log_b(x) | ln(x) / ln(b) | Base-change formula |
Key values, base conversions, and rules for quick reference.
Frequently asked questions
What is the difference between ln and log?
ln is the natural logarithm with base e (about 2.71828), while "log" most often means base 10 (the common logarithm) in everyday and engineering contexts, and base e in pure mathematics and many software languages. They differ by a constant factor: ln(x) = log10(x) * ln(10), approximately 2.302585 * log10(x).
What are ln(1) and ln(e)?
ln(1) is exactly 0, because e^0 = 1. ln(e) is exactly 1, because e^1 = e. These two anchor points are the most important values to memorize.
Why can I not take the natural log of a negative number?
No real power of e produces a negative result or zero, so ln(x) has no real value for x at or below zero. Negative inputs only have logarithms in the complex number system, where ln(-1) = i*pi. This calculator covers real numbers only.
How do I reverse a natural log (find x given ln(x))?
Select Inverse mode and enter your known ln value. The calculator computes x = e^y, which is the exponential function and the exact inverse of the natural log. For example, if ln(x) = 2, then x = e^2 ≈ 7.389.
How do I convert between log bases?
Use the base-change formula: log_b(x) = ln(x) / ln(b). Select Log base b mode, enter x and the desired base, and the calculator divides the two natural logs to give log_b(x). Common bases: 10 gives log10, 2 gives log2 (bits), and 0.5 or other fractions are also valid bases (though unusual).
What is the Rule of 70 and where does ln come from?
The Rule of 70 says doubling time ≈ 70 / (rate in percent). It comes from the exact formula T = ln(2) / r, where ln(2) ≈ 0.6931 and multiplying by 100 gives about 69.3, rounded to 70 for easy mental arithmetic. The rule is accurate within 1% for rates under about 15% per year.
How is the natural log used in radioactive decay?
The decay law is N(t) = N0 * e^(-lambda*t), where lambda is the decay constant. Taking the natural log of both sides: ln(N/N0) = -lambda*t. The half-life T_{1/2} = ln(2) / lambda. Enter the decay rate (lambda * 100 as a percentage) in Half-life mode to find T_{1/2} directly.