Find the natural logarithm of a number.
The natural log uses Euler's number e as its base, and appears throughout calculus and compound growth formulas.
ln(x): Logarithm base e (≈2.71828)
A natural log calculator finds ln x, the logarithm of x to the base e, where e is approximately 2.71828. It answers the question: to what power must e be raised to give this number? For x = 2, ln 2 ≈ 0.693, a number that turns up whenever something doubles under continuous growth.
The natural logarithm is the default logarithm in calculus, physics, biology and finance. Class 12 students meet it when integrating 1/x, and it drives formulas for radioactive half-life, population growth, capacitor discharge and continuously compounded interest. In finance, log returns are often used because they add neatly across time. The calculator accepts any positive number.
1. Check that x is positive; ln is not defined for zero or negative numbers.
2. If x is a known power of e, read it off: ln(eᵏ) = k, so ln e = 1 and ln 1 = 0.
3. Otherwise convert from base 10: ln x = log₁₀ x ÷ log₁₀ e, which is about 2.302585 × log₁₀ x.
4. Use log tables or a calculator for log₁₀ x.
5. Multiply to get ln x.
6. Check by computing e raised to your answer; it should give x.
Among all exponential functions bˣ, the one with base e has the special property that its rate of change equals its own value. Correspondingly, the slope of ln x at any point is 1 ÷ x. So ln x is the area under the curve 1/t from 1 to x, which is how it is introduced in calculus. This clean derivative is why growth and decay equations are written with e and ln, and why other bases would introduce awkward constant factors.
If a quantity grows continuously at rate r per year, its value after t years is P × e^(rt). To find when it doubles, set e^(rt) = 2 and take natural logs: rt = ln 2, so t = ln 2 ÷ r ≈ 0.693 ÷ r. At 7% continuous growth, doubling takes about 9.9 years. This is the origin of the rule of 70 and, with adjustment for yearly compounding, the rule of 72 used in personal finance. Half-life follows the same formula with decay.
Natural and common logarithms differ only by a constant factor: ln x = ln 10 × log₁₀ x ≈ 2.302585 × log₁₀ x. So any fact about one carries over to the other. Both are zero at x = 1, both are negative between 0 and 1, and both grow slowly for large x. Chemistry often uses log₁₀ for pH, while physics and calculus use ln. Using the wrong one introduces an error by a factor of about 2.3.
Shreya, studying for her Class 12 exams, needs ln 2 to find how long a bacterial culture growing continuously at a fixed rate takes to double.
Answer: ln(x) 0.693147
Using log₁₀ where a formula specifies ln.
Trying to take ln of zero or a negative number.
Writing ln(a + b) = ln a + ln b; the rule works only for products.
Forgetting that ln of a number between 0 and 1 is negative.
Doubling time and half-life calculations in biology, chemistry and physics.
Continuously compounded interest and log returns in finance.
Integration and differentiation in Class 12 calculus.
Capacitor charging and discharging in electronics.
Entropy and information measures in physics and computer science.
Why is e used as a base at all?
e comes up naturally in continuous growth. It is the base where the logarithm's rate of change is simplest, which is why it appears so often in calculus and finance.