LnCalc

Natural Logarithm Calculator

Enter a number — get ln, log₁₀, and log₂ with the change-of-base formula shown.

x (positive number)

ln(x)

2.303

log₁₀(x): 1
log₂(x): 3.322

What you entered

x: 10

1
Natural logarithm: ln(x)
ln(10)= 2.3026
2
Common log: log₁₀(x) = ln(x) / ln(10)
2.3026 / 2.3026= 1
3
Binary log: log₂(x) = ln(x) / ln(2)
2.3026 / 0.6931= 3.3219

Result

ln(x): 2.303

The natural logarithm of 10 is 2.303. In other bases: log₁₀(10) = 1 and log₂(10) = 3.322.

Natural log alongside log base 10 and base 2

The natural log uses base e, about 2.71828. All three logarithms of the same number are shown for comparison.

x	ln x	log10 x	log2 x
1	0.000000	0.000000	0.000000
2	0.693147	0.301030	1.000000
e	1.000000	0.434294	1.442695
10	2.302585	1.000000	3.321928
100	4.605170	2.000000	6.643856
1000	6.907755	3.000000	9.965784

Each base gives 1 for its own value - ln e, log10 of 10 and log2 of 2 all equal exactly 1 - and every base gives 0 at x = 1. ln 2 at 0.693147 is worth recognising: it is the constant in every exponential decay half-life formula, and dividing it by a growth rate gives the exact doubling time that the Rule of 72 approximates. The three columns are all proportional to each other, so converting between bases is just a fixed multiplication.

Why natural log uses e

The natural logarithm ln(x) uses base e ≈ 2.71828, Euler's number. It arises naturally in continuous growth/decay: if something grows at 100% per year compounded continuously, it multiplies by e each year. ln(x) answers 'what continuous rate for how long gives factor x?'

Change of base formula

Any logarithm can be converted: log_b(x) = ln(x)/ln(b). So log₁₀(x) = ln(x)/ln(10) and log₂(x) = ln(x)/ln(2). This is useful because most programming languages and calculators provide ln directly.

Frequently asked questions

An investment doubled in 8 years — what was the continuous annual growth rate?

If it doubled, the growth factor is 2. For continuous growth: 2 = e^(r×8), so r = ln(2)/8 ≈ 0.6931/8 ≈ 0.0866 or about 8.66% per year. The natural log converts a growth factor directly into a continuous rate.

How is ln different from log₁₀?

ln uses base e ≈ 2.718 (natural for continuous processes); log₁₀ uses base 10 (natural for orders of magnitude). ln(100) ≈ 4.605 while log₁₀(100) = 2. Calculus and most science use ln; decibels and Richter scales use log₁₀. They're related: ln(x) = log₁₀(x) × ln(10) ≈ log₁₀(x) × 2.3026.

Why does my programming language use 'log' for the natural log instead of 'ln'?

Most programming languages (Python, JavaScript, C) use Math.log() for the natural logarithm. This follows a mathematical convention where 'log' without a base defaults to ln in higher math. For base-10, they provide Math.log10(). The logarithm calculator handles all three common bases.