ln(1000), sum of logarithms
- Input
- x=1000
- Expected output
- ln(1000) ≈ 6.907755279
Matches 3×ln(10)=3×2.302585093=6.907755279, confirming the property ln(10³)=3×ln(10).
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calculate natural logarithm ln
ln(x) is the exponent e≈2.718281828 must be raised to in order to reach x. It is the natural base of differential calculus and of continuous growth and decay modeling, where ln(2)≈0.693147 keeps showing up as the constant behind doubling time.
Matches 3×ln(10)=3×2.302585093=6.907755279, confirming the property ln(10³)=3×ln(10).
Used in t=ln(2)/r: at 5% a year (r=0.05), a quantity takes ≈13.86 years to double.
It is the exact negative of ln(2), because 0.5 is the reciprocal of 2 and ln(1/x)=−ln(x).
log_b(x) = y means b^y = x, so a logarithm is simply an exponent, the power the base must be raised to. It is the inverse operation of exponentiation. For example, log₂(8) = 3 because 2³ = 8, and log₁₀(100) = 2 because 10² = 100.
Because e⁰=1 is the reference point: any x below 1 is only reached by a negative exponent, since positive powers of e≈2.718281828 always result in values above 1. ln(0.5)≈−0.693147 is a direct example of that rule.
The formula is t=ln(2)/r for continuous growth at rate r. ln(2)≈0.693147 because e raised to 0.693147 results in approximately 2; for a 5% annual rate, the doubling time is 0.693147/0.05≈13.86 years.
No. The calculator requires x>0 because no real exponent y makes e raised to y result in zero or a negative number: e^y is always positive for any real y, so no real ln exists for zero or a negative number.
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