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Natural Log Calculator

Enter a positive number to compute its natural log.

How it is calculated

For a positive number x the tool returns the power to which a base must be raised to give x, in three bases. The natural log is ln(x), base e ≈ 2.71828. The common log is log₁₀(x) = ln(x) ÷ ln(10) and the binary log is log₂(x) = ln(x) ÷ ln(2), with ln(10) ≈ 2.302585 and ln(2) ≈ 0.693147. All three are shown to 6 decimals. The logarithm exists only for numbers above zero, so 0 or a negative number gives no result. Values between 0 and 1 give negative logs, and x = 1 gives 0 in every base. Because the three results differ only by a constant factor, once you know one you can find the others by multiplying or dividing by ln(10) or ln(2).

Worked example

Take x = 50. Since 50 = 100 ÷ 2, ln(50) = ln(100) − ln(2) = 4.605170 − 0.693147 = 3.912023, so e raised to about 3.912 is 50. The base-10 log is 3.912023 ÷ 2.302585 = 1.698970, meaning 101.69897 = 50. The base-2 log is 3.912023 ÷ 0.693147 = 5.643856, meaning 25.643856 = 50, a little over 5.6 doublings starting from 1.

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Common mistakes

  • Taking the log of zero or a negative number. These have no real logarithm, so the tool returns nothing; check whether you meant the absolute value.
  • Confusing ln with log. Many calculators and texts use "log" for base 10, others for base e; here ln is base e and log₁₀ is shown on its own line.
  • Expecting ln(a + b) to equal ln(a) + ln(b). Logs turn products into sums, so only ln(a × b) = ln(a) + ln(b); there is no simple rule for a sum.
  • Treating a result below 1 as a mistake. For x between 0 and 1 the log is negative (ln 0.5 = −0.693147), which is correct.
  • Rounding too early in a chain of calculations. The display stops at 6 decimals, so for later steps (such as a doubling-time formula) keep the unrounded value if you need accuracy.

About this calculator

The natural logarithm, written ln(x), answers the question “e to what power gives x?” where e ≈ 2.71828 is Euler's number. It's the inverse of the exponential function eˣ, and it shows up naturally wherever continuous growth or decay is involved — compound interest compounded continuously, radioactive decay, and the growth curves in calculus and differential equations.

Because different fields prefer different bases, this calculator also converts to base-10 (common, used in engineering and pH/decibel scales) and base-2 (used throughout computer science, e.g., for measuring algorithmic complexity) using the change-of-base formula, log_b(x) = ln(x) / ln(b). Scientists, engineers, and students use ln both as a standalone tool and as a building block inside larger formulas like the Black-Scholes option pricing model.

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Comments (1)

  • Editorial team

    Logarithms turn growth questions into division. To find how long something takes to double at a steady growth rate, divide ln 2 (0.693147) by ln of one plus the rate: at 7% a year, 0.693147 divided by ln 1.07 is about 10.2 years. The three bases always scale each other, so ln 100 is 4.605170 and log10 100 is exactly 2. What equation are you solving with it?

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