# Logarithms chronologically before algebra

Do any textbooks or (somewhat?) standard curricula introduce logarithms and their applications in arithmetic without assuming the students know any algebra?

(I do not mean just the use of logarithms for doing arithmetic, but rather, understanding of logarithms.)

• I've seen examples of this, but as far as I can recall now, only in popularizations and semi-technical literature, such as Isaac Asimov's Realm of Numbers, not in the type of literature you're asking about. Nov 16 '20 at 11:28
• Before 1975 there had to be many, many people who used slide rules but didn't know any algebra.
– user507
Nov 16 '20 at 16:48
• It surprises me a bit that it isn't in the Common Core for middle school, because you could teach that roots and logarithms are both inverses of the exponential operation since it is not commutative. But evidently logarithms aren't covered until Algebra 2 (in the US). Nov 16 '20 at 16:48

Chemistry books through the 80s, at least, taught the use of log tables for doing arithmetic calculations, without stressing the teaching of logs as inverse power functions. Purely as a sort of table based slide rule. You can make hs physics problems easy with round numbers but it's pretty much impossible with chemistry, given molar masses, gas laws, moles, etc. etc.

The need for this went away after mid to late 70s when hand calculators had become cheap and ubiquitous in chemistry classes. But I still remember working some calculations with log tables in the late 70s, early 80s in chem classes. The AP Chem exam had log tables, as aids, into the 90s, but it was just inertia. Scientific calculators were the norm.

Not precisely same, but similar is the use of tables for future and present time value of money in engineering economics, rather than calculating the details algebraically. See for example the chapter within the EIT reference manual by Lindberg.

• All this is true, but I had in mind actual understanding of logarithms, rather than mere use of logarithms for doing arithmetic. Nov 22 '20 at 22:14
• @MichaelHardy You could edit the question to clarify/emphasize. Nov 23 '20 at 7:50
• @MichaelHardy: What makes you think that they did those calculations without "actual understanding" of logarithms? If you understand that logs add when you multiply, etc., then I think you do have an actual understanding of what they are.
– user507
Nov 23 '20 at 19:45
• It would be interesting to understand how (or if) this meshed with the math curriculum in a year like 1960. It would also be interesting to know whether this was done in both high school and college chem textbooks.
– user507
Nov 23 '20 at 20:00
• I checked a copy of PSSC Physics, a high school book from 1960 meant for talented and well-prepared students. It has trig tables but no log tables. There is no entry in the index under "logarithm." So I wonder if this was done in college-level physics and chem texts, since college students were presumed to have formally encountered logs already.
– user507
Nov 23 '20 at 20:12

Generally, it's presumed students know about directed numbers (which may or may not be considered algebra), because otherwise they can't use logarithms of numbers less than 1.

But apart from that proviso, the answer is that this was absolutely the case in some countries, before calculators replaced logarithm tables. For students aiming for O-levels in England (taken at age 16 by roughly the top 25% in overall academic ability), logarithms were typically introduced at age 12-14, with an emphasis on practical calculations but little on theory. While it is also true that students were acquainted with some algebra at this age, logarithms were generally presented only numerically at this stage.

For example, if you look at the old O-level syllabi of one examining board, you'll see that logarithm tables were used in exams as late as 1984, and they were explicitly part of the arithmetic syllabus as late as 1974.

For examples of textbooks with early introduction of logarithms, you could have a look at General Arithmetic for Schools by Durell, or Arithmetic, Part II by Siddons, Snell and Lockwood.

• The problem with this is that your answer addressed the use of tables, rather than understanding of logarithms of the kind that persists if table become obsolete. Dec 3 '20 at 6:30
• I think the laws $\log xy = \log x + \log y$ and $\log x^a = a \log x$ would have been understood, at least when $a$ is an integer or the reciprocal of a positive integer. For general $a$ or for bases other than $10$, this understanding would have come later, with algebra. Depending on the particular textbook, an understanding of the inverse relationship between the logarithm function and $10^x$, as well as their graphs, might form part of what was taught to 12-14 year-olds. What kinds of understanding do you have in mind that would be compatible with not knowing algebra?
– Dave
Dec 3 '20 at 7:56
• For example, in Durell's arithmetic you find the following problems: Find $x$ if $3^x = 1000$. Simplify without using tables: $\log 9/\log 3$, $\log \sqrt{7}/\log 7$, $\log 6\frac{1}{4} + \log 1\frac{3}{5}$. Find the least integral value of $n$ for which $(0.95)^n$ is less than $0.1$. These all require some degree of understanding of the abstract properties of logarithms. You can be the judge of whether this meets your criteria.
– Dave
Dec 3 '20 at 8:10
• Look at the "powers of 10" book or a graphs that have the $y$-axis on logarithmic scales. Dec 4 '20 at 3:08
• Graphs with logarithmic scales would have been discussed in the context of algebra, since in many cases the point would be to determine the constants in a relation $y = ax^b$ from experimental data. This hardly seems sensible without knowing any algebra.
– Dave
Dec 4 '20 at 4:33

One example where you might expose students to the idea of logarithms prior to algebra might be through log-scaled graphs, such as this simple example below:

Students might not have the technical language to express what is going on in the graph to the right, but I think that it is within the realm of their comprehension. Especially with the advent of COVID, it might be useful to expose pre-algebra students to this kind of growth and model so that they might have a better understanding of why disease transmission is something that needs to be taken seriously.