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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 20900 21609 90235 30436 99418 49146 31409 34317 38143 64054 62531 52096 18369 08887 07016 76839 64243 78140 59271 45635 49061 30310 72085 10383 75051 01157 47704 17189 86106 87396 96552 12671 54688 95703 50354 02123 40784 98193 34321 06817 01210 05627 88023 51930 33224 74501 58539 04730 41995 77770 93503 66041 69973 29725 08868 76966 40355 57071 62268 44716 25607 98826 51787 13419 51246 65201 03059 21236 67719 43252 78675 39855 89448 96970 96409 75459 18569 56380 23637 01621 12047 74272 28364 89613 42251 64450 78182 44235 29486 36372 14174 02388 93441 24796 35743 70263 75529 44483 37998 01612 54922 78509 25778 25620 92622 64832 62779 33386 56648 16277 25164 01910 59004 91644 99828 93150 56604 72580 27786 31864 15519 56532 44258 69829 46959 30801 91529 87211 72556 34754 63964 47910 14590 40905 86298 49679 12874 06870 50489 58586 71747 98546 67757 57320 56812 88459 20541 33405 39220 00113 78630 09455 60688 16674 00169 84205 58040 33637 95376 45203 04024 32256 61352 78369 51177 88386 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44013 39762 20967 49454 18540 71184 46433 94699 01626 98351 60784 89245 14058 94094 63952 67807 35457 97003 07051 16368 25194 87701 18976 40028 27648 41416 05872 06184 18529 71891 54019 68825 32893 09149 66534 57535 71427 31848 20163 84644 83249 90378 86069 00807 27093 27673 12758 19665 63941 14896 17168 32980 45513 97295 06687 60474 09154 20428 42999 35410 25829 11350 22416 90769 43166 85742 42522 50902 69390 34814 85645 13030 69925 19959 04363 84028 42926 74125 73422 44776 55841 77886 17173 72654 62085 49829 44989 46787 35092 95816 52632 07225 89923 68768 45701 78230 38096 56788 31122 89305 80914 05726 10865 88484 58731 01658 15116 75333 27674 88701 48291 67419 70151 25597 82572 70740 64318 08601 42814 90241 46780 47232 75976 84269 63393 57735 42930 18673 94397 16388 61176 42090 04068 66339 88568 41681 00387 23892 14483 17607 01166 84503 88721 23643 67043 31409 11557 33280 18297 79887 36590 91665 96124 02021 77855 88548 76176 16198 93707 94380 05666 33648 84365 08914 48055 71039 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.

Please see my example below:

We’ll continue our A to Z of Excel Functions soon. Keep checking back – there’s a new blog post every business day.

A full page of the function articles can be found here.

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