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In my understanding, every operation we do in $\log_{10}$ can be done in the natural logarithm itself and it should be better because mathematical integrals naturally give out expressions involving the natural logarithm function.

However, it seems that in my chemistry books there are elaborate schemes set up to avoid this and use the base ten logarithm instead. For example, we define $\mathrm{pH}$ as the negative of log base ten of hydrogen concentration instead of choosing the natural logarithm.

What are the historical reasons in the study of chemistry which caused this preference?

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    $\begingroup$ Largely due to comparing things that span many orders of magnitude. pH of 6 and pH of 8 is otherwise a factor of 100: try plotting that. As a geologist, I'll direct you to the inconvenience of base 2 for sorting grain sizes in sediments. It's all about who did it first AND how convenient it is for all parties concerned. $\endgroup$
    – Todd Minehardt
    Jun 7 '21 at 1:42
  • $\begingroup$ If we had 8 digits in our hands instead of the usual 10, we would use a base-8 system (most likely). Consider also the odd situation of computer scientists who choose to speak of kB, MB, GB, TB .... when in fact they mean the nearest binary number. It's ingrained in our way of counting learned from childhood. Perhaps kindergartens should teach to count using the binary system (base-2). Base-e (natural log) is then something else. You are not even using rational numbers. Much harder to rationalize its use. $\endgroup$
    – Buck Thorn
    Jun 7 '21 at 6:02
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    $\begingroup$ As MFarooqs answer explains, a unit pH difference is easy to conceptualize: a factor of 10 difference in concentration. pH 2 difference = factor of 100, etc. Easy! Now think about ln scale: -lnH difference of 1 means difference of e in concentration; a difference of 2 means $e^2$... uhm, what ? Note we already use a base-10 (decimal) system. Why deviate from that (like the Imperialists)? My favorite unit in this context has to be the decibel. $\endgroup$
    – Buck Thorn
    Jun 7 '21 at 6:07
  • $\begingroup$ @BuckThorn Though in French, e.g., shopping for a thumb drive, you often read ko, Mo, Go, To where «o» stands for «octet» in lieu of kB, MB, GB (example) recalling that there is some base of eight in play ... $\endgroup$
    – Buttonwood
    Jun 7 '21 at 15:09
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    $\begingroup$ There are cases where 2.303 (ln(10)) coefficient has to be used if log10 is used instead of ln. There are cases where 1/2.303 coefficient would be used if ln was used instead of log10. All depends if the scenario has "natural" or decimal nature. $\endgroup$
    – Poutnik
    Jun 25 '21 at 11:30
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All fields of science (biology, chemistry, physics, and geology) use both natural and base-10 logarithms.

Is there a distinguishing characteristic that (typically) determines when one is used vs. the other? Yes: There is a class of applications where the natural log is used, and a qualtitatively different class of applications where the base-10 log is used.

Natural logarithms are used when describing physical processes whose underlying mathematics are exponential (specifically, base-$e$ exponential, which is commonly referred to simply as "exponential"). Examples:

Biology: Population growth

Chemistry: First-order rate laws

Chemistry and Physics: Nuclear decay

Exponentials are commonplace because they result from a frequently-seen type of physical process: Those where the rate at which the quantity of an item changes is proportional to how much of that item you have (i.e., where the derviative of a function equals its value):

$$\frac{dy}{dx} = y \implies \frac{dy}{y} = dx \implies \ln y = x \implies y = e^x$$

Natural logs are used to linearize these functions.

In summary, natural logs appear when mathematically characterizing physical processes that follow a very common functional form: the exponential.

[In the comments, Peter Mortensen noted that base-2 may be more natural for nuclear decay than base-$e$. The underlying physics is exponential, but base-2 becomes convenient specifically if you want to work in terms of half-life—which is, of course, a standard way of specifying nuclear decay rates: $N(t) = N_o e^{-\lambda t} \implies N(t) = N_o e^{-t\ln 2/\tau} \implies N(t) = N_o 2^{-t/\tau}$, where $\tau$ = half-life.]

By contrast, base-10 logs are used when deploying a measurement scale that needs to cover a wide range of values. Examples include the pH scale in chemistry, and the Richter scale in geology. None of these scales are mathematically tied to any underlying physical process. They are purely descriptive. E.g., the Richter scale is based on the base-10 logarithm of the amplitude of seismogram waveforms.

When creating such purely descriptive scales it's typically considered much more convenient and intuitive to base them on powers of 10 rather than powers of $e$. Hence in converting the values of these phenomena to their respective scales, the base-10 logarithm is typically used rather than the base-e ("natural") logarithm.

So no, it's not the case that "every operation...should be better [with the natural log]"; nor are chemists concocting "elaborate schemes" to avoid using it. Rather, chemists (and scientists in all other fields) readily use both the natural log and $\log_{10}$, depending on what makes sense.

[Note: I'm not saying that $\ce{[H^+]}$ or seismogram waveform amplitude couldn't be tied to some underlying physical process; I'm simply saying these scales don't do that.]

The base-10 logarithm is also often used purely for presentational purposes, when showing data covering a wide range. Here's an example from ATLAS Collaboration, Probing Dark Matter with the Higgs boson, April 2020, https://atlas.cern/updates/briefing/probing-dark-matter-higgs-boson:

enter image description here

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    $\begingroup$ To sum it up: Decimal numbers use base 10, which naturally propagates where there are no (significant) other concerns. $\endgroup$ Jun 7 '21 at 10:12
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    $\begingroup$ Isn't base 2 more natural (no pun intended) for nuclear decay (half-life, no need for ln(2))? (Not a rhetorical question.) $\endgroup$ Jun 7 '21 at 10:29
  • $\begingroup$ @PeterMortensen Good point. The exponential function corresponds directly to the physics of nuclear decay. However, yes, it is natural (and usual) to think in terms of half-lives and, specifically for that reason, it is not uncommon to see nuclear decay re-expressed in base-2, i.e.: $N(t) = N_o e^{-\lambda t} \implies N(t) = N_o e^{-\ln 2 t/\tau} \implies N(t) = N_o 2^{-t/\tau}$, where $\tau$ = half-life. I'll edit my post to add that. But my broader point remains: those working in the field use what makes sense for the intended use and, if applicable, the underlying physical process. $\endgroup$
    – theorist
    Jun 8 '21 at 2:30
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I am sure you are mathematically gifted, and that's why you sensed this trend in an experimental science like chemistry. It seems that your textbook author(s) put an unnecessary emphasis on avoiding natural logarithms. The quick answer is that depending on the experimental situation, one can use either the natural logarithm or Briggsian logarithm. Chemists do not avoid natural logarithms. For example, in chemical rate laws or radioactive decay, the natural logarithm is the natural way to go.

However, when it comes to solution preparation, base 10 is experimentally feasible and of course very convenient even for mental calculations. When chemists like to prepare solutions, often the concentration is a simple multiple of 10.

Our South Asian readers will immediately recognize the normality notation often labelled on acids or bases as 1 N, N/10, N/100 solutions. Normality is equal to gram equivalents/ L. In the 18th and 19th century, this unit was in vogue. It makes solution preparation and dilution easier. Recall that burettes or graduate pipets are all scaled like 0.1, 0.2, 0.3 mL volumes and so on rather than multiples of natural logarithm base $\mathrm e$. You will notice natural logarithm avoidance in the pH scale as well as the Nernst equation, because concentration terms are involved. As one reader pointed out Nernst equation is typically found with natural logs on Wikipedia. I looked the original 100 year old text of Nernst, and he indeed used natural logs. My bias is toward analytical chemistry and then consulted a classic on electroanalytical chemistry text (Lingane: Electroanalytical Chemistry), there the author after doing all the derivations switches to Briggsian log. The reason that one of the most reliable methods to determine pH was electrochemical methods. All your pH meters rely on that equation i.e., Nernst equation with Briggsian log. So this choice, is again dependent on the needs of the researcher.

Sørensen is popularly called the originator of pH concept. However, the hydrogen ion concentration idea was well known in his time. I checked Sørensen's original essay (Biochemische Zeitschrift. v.21 1909, freely available from Hathi Trust), and all his concentration units are in normality. As you can see for the pH, the scale goes like the picture below (Ref: Italian Journal of Medicine (2011) 5, 147—155). Now imagine setting up a natural logarithm concentration scale. This is not experimentally feasible. The numbers look "friendlier" with base 10 logarithms.

$$ \begin{array}{rll} \hline \mathrm{pH} & [\ce{H+}]/\pu{g Eq L^-1} & [\ce{H+}]/\pu{g Eq L^-1} \\ \hline 0 & 1 & 1.0 \\ 1 & 0.1 & \pu{1.0E-1} \\ 2 & 0.01 & \pu{1.0E-2} \\ 3 & 0.001 & \pu{1.0E-3} \\ 4 & 0.0001 & \pu{1.0E-4} \\ 5 & 0.00001 & \pu{1.0E-5} \\ 6 & 0.000001 & \pu{1.0E-6} \\ 7 & 0.0000001 & \pu{1.0E-7} \\ 8 & 0.00000001 & \pu{1.0E-8} \\ 9 & 0.000000001 & \pu{1.0E-9} \\ 10 & 0.0000000001 & \pu{1.0E-10} \\ 11 & 0.00000000001 & \pu{1.0E-11} \\ 12 & 0.000000000001 & \pu{1.0E-12} \\ 13 & 0.0000000000001 & \pu{1.0E-13} \\ 14 & 0.00000000000001 & \pu{1.0E-14} \\ \hline \end{array} $$

And students of the 1970 and early 1990s era will recall that base 10 logarithm tables or a slide rule were part and parcel of every student who studied mathematics and natural sciences. We used to have printed copies of base 10 logarithm tables in school in mathematics classes.

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    $\begingroup$ Re "And students of the 1970 and early 1990s era will recall that base 10 log tables or a slide rule were part and parcel of every student who studied mathematics and natural sciences". Where? I have never encountered them (only outside of the education system - my parent's generation knew them - I have my father's slide rule). $\endgroup$ Jun 7 '21 at 10:11
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    $\begingroup$ I doubt the "early 1990" era for slide rules. I was born in 1966, and at my school, we were the first 7-graders - so this would have been 1979 - who got pocket calculators, a friend who was 1 year older had still been taught to use slide rules. $\endgroup$ Jun 7 '21 at 10:48
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    $\begingroup$ But the Wikipedia article on the Nernst equation has way more $\ln$'s than $\log_{10}$'s; and the $\log_{10}$-based equations require the awkward conversion factor of $\ln 10 = 2.303$. After all at the end of the day it boils down to thermodynamics which uses $\ln$: $\Delta G = \Delta G^\circ + RT\ln Q$, and then to $\mu = \mu^\circ + RT\ln a_i$. In general I agree, but I think a better example could be chosen. $\endgroup$
    – orthocresol
    Jun 7 '21 at 13:29
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    $\begingroup$ @PeterMortensen, The log table were purely academic excercises in schools (not colleges or universities) in South Asia until the late 90s. $\endgroup$
    – M. Farooq
    Jun 7 '21 at 13:33
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    $\begingroup$ @orthocresol, Good observation, I updated the post. $\endgroup$
    – M. Farooq
    Jun 7 '21 at 14:16
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I'm guessing, of course, but my answer is: graph paper.

Every proportionality produces a plot on cartesian graph paper that is a straight line. And, every power law produces a plot on log-log paper that is a straight line. Every exponential (exponentials are faster-rising than any power law) can be made to describe a straight line on a plot using semilog paper.

So, part of yesteryear's equipment any experimenter needed was a selection of graph papers to hand-enter data points onto; that was the way to make a curve fit for those (fairly common) sorts of relationships. I'm sure Pierre Curie used log-log paper to determine the power law for magnetization diminution near the Curie temperature, for instance. How else, before easy computer access, would he have found that constant (about 1/2)?

And, to enter a decimal number onto logarithmic scale, you are using axes for your graph that have a small-integer-ratio that makes the log(base 10) trivial to read off. One still hears lots of '3 dB per octave' chatter, in some circles, which is a reference to the 'octave', a unit interval of a time or frequency scale, which 'unit' is in the logarithm base 2.

While some use is found for base 10 and base 2 scales, I've never seen a graph paper log 'cycle' which was a factor of 'e'.

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In addition to the other answers:
In the old days, log calculations were not simply a click or press on a computer / calculator, but needed to be looked up in paper tables - which in turn took man-years of manual calculations to produce, and therefore were expensive. Why buy another such booklet for ln, if you already had the one for log10 (and knew a lot of logs by heart)?

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