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# A to Z of Excel Functions: The LN Function

11 October 2021

Welcome back to our regular A to Z of Excel Functions blog. Today we look at the LN function.

The LN function

So you get bored one afternoon and decide it’s time to sum the reciprocals of all of the factorial numbers:

This constant is known as Euler’s number (e) and is equal to the limit of:

as n approaches infinity, an expression that arises frequently in the study of compound interest. It is equal to

e = 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69995 95749 66967 62772 40766 30353 54759 45713 82178 52516 64274 27466 39193 20030 59921 81741 35966 29043 57290 03342 95260 59563 07381 32328 62794 34907 63233 82988 07531 95251 01901 15738 34187 93070 21540 89149 93488 41675 09244 76146 06680 82264 80016 84774 11853 74234 54424 37107 53907 77449 92069 55170 27618 38606 26133 13845 83000 75204 49338 26560 29760 67371 13200 70932 87091 27443 74704 72306 96977 20931 01416 92836 81902 55151 08657 46377 21112 52389 78442 50569 53696 77078 54499 69967 94686 44549 05987 93163 68892 30098 79312 77361 78215 42499 92295 76351 48220 82698 95193 66803 31825 28869 39849 64651 05820 93923 98294 88793 32036 25094 43117 30123 81970 68416 14039 70198 37679 32068 32823 76464 80429 53118 02328 78250 98194 55815 30175 67173 61332 06981 12509 96181 88159 30416 90351 59888 85193 45807 27386 67385 89422 87922 84998 92086 80582 57492 79610 48419 84443 63463 24496 84875 60233 62482 70419 78623 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76521 46960 27662 58359 90519 87042 30017 94655 3679 ... courtesy of University of Utah

You should learn this.  It will make you much more desirable at parties.

Like πe is transcendental: it is not a root of any non-zero polynomial with rational coefficients, but it is a very powerful number in the world of mathematics.  Bizarrely, this constant was not discovered by Swiss mathematician Leonhard Euler, but rather by his compatriot, Jacob Bernoulli, while studying compound interest.  If it had been named after the latter, blog (rather than loge) might mean something entirely different…

The natural logarithm of a number is its logarithm to the base of said constant e.  The natural logarithm of x is generally written as ln x (where n is after John Napier, the discoverer of logarithms and the “natural” base, e, is also referred to as Napier’s constant) or less frequently, loge x.

The natural logarithm of x is the power to which e would have to be raised to equal x.  For example, ln 7.5 is 2.0149..., because e2.0149... = 7.5.  The natural logarithm of e itself, ln e, is 1, because e1 = e, whilst the natural logarithm of 1 is 0, since e0 = 1.

The function slowly grows to positive infinity as x increases and slowly goes to negative infinity as x approaches 0 ("slowly" as compared to any power law of x); the y-axis is an asymptote, viz.

The LN function employs the following syntax to operate:

LN(number)

The LN function has the following arguments:

• number: this is required and represents the positive real number for which you want the natural logarithm.

It should be further noted that:

• LN is the inverse of EXP, the natural exponential of number.