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I'm trying to help someone pass their final exam (analysis of functions) so they can graduate high school and move on to college. (Not a teacher, just another student, currently in high level math)

They are currently struggling with many concepts, most recently including ln.

She is very frustrated because she feels like she needs to know what ln is and how to find it by hand and how to do everything by hand. If she uses a calculator, she feels like she doesn't understand and isn't learning the math. (In our specific problem, I told her she needed a calculator to enter ln(5.4) and take that divided by 8 for her answer.) Her teacher is far less than helpful (I'm not convinced her teacher even understands it), leaving her more frustrated.

I explained to her that ln is a function, like an oven where you bake a cake. You put something in, and depending what you put in, you get something else out. The calculator is just the oven, and ln is just a recipe. You are still the one doing the baking.

She asked what people did before calculators and I told her that before calculators, you used a slide rule, and before that, you used a log table. Nobody ever did them by hand. I compared it to deriving pi by hand; possible, but pointless.

She wanted to know where ln comes from and what it meant, so I met her with this wonderful ms paint diagram:

My fabulous ms paint skills

This seemed to help her understand finally where ln comes from, after I also explained that ln is just log base e.

But she was still frustrated that she couldn't do it by hand and didn't know how to find the area. I warned her that this is literally second semester college calculus but she seemed to need to know, so I briefly explained to her the infinite series used to calculate it as simply as possible, which I think she actually understood. She asked why she couldn't do it by hand, and I told her the obnoxious number of iterations she would have to do.

I think I finally got the concept of ln through to her, but I fear she will have similar problems with the rest of her work with logs, and upcoming sigma notation (for many iterations, not wanting to do it in the n-spire), and other things. How can I convince her that the mechanics of the computations aren't what are important here?

I tried explaining that these questions will be answered in higher math, but seems unable (unwilling?) to grasp that understanding the chemistry of baking isn't needed to know how to make a good cake.

It is worse because, asking why she needs to know this, I really don't have a good answer. Most of math is practical, but I was always confused as to why we teach lns and e without the proper calculus background to really understand them.

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    $\begingroup$ Could you post the question that she was stuck with? $\ln$ and $e $ crop up in many varied situations and I think it'd help. I understand the frustration you have but blaming the teacher isn't helpful for anybody. $\endgroup$ – Karl May 17 '15 at 18:00
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    $\begingroup$ I realized blaming the teacher isn't helping anyone; just venting some frustration. From what I hear from other, reliable students, the teacher just throws up a problem, does it, and then sits there at her desk doing stuff and gets upset when people ask questions. Not helpful for anyone. But anyways, the problem was solve for four decimals: $e^{8x} = 5.4$ $\endgroup$ – Bassinator May 17 '15 at 18:45
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    $\begingroup$ It is a shame that she isn't given the opportunity to give the exact answer of $\displaystyle\frac{\ln (5.4)}{8}$. $\endgroup$ – JP McCarthy May 17 '15 at 19:45
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    $\begingroup$ Her desire to understand rather than recite by rote is commendable, and you should make that clear. You might try telling her that in the days before electronic calculators people were expected to look the answers up in books of tables. Here is a page showing how the "Godfrey and Siddons" 4 figure tables were used: gwydir.demon.co.uk/jo/numbers/machine/log.htm $\endgroup$ – Paul Johnson May 17 '15 at 21:52
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    $\begingroup$ It should be mentioned that "Nobody ever did them by hand" is in fact false: log tables did not popped up from nowhere, they where actually computed by hand. At some point in time, I think that computing a good log table could be one's great life achievement; this should explain why we do not do that know that we don't have to. $\endgroup$ – Benoît Kloeckner May 19 '15 at 14:16
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A possible source of the need to do it by hand

The problem might be to do with the examples she saw of logs and exponentials before she saw $\ln$.

Usually, when logs are introduced, they are introduced as the inverse of exponential functions (which is what they are of course). However, almost all of the examples and exercises are with whole numbers. For example, the statement $\log_2 (8) = 3$ is declared to be true because $2^3 = 8$. Also, examples of figuring out logs will be with whole numbers too, and will boil down to guessing what power of a whole number produces the answer. For example, finding $\log_3 (81)$ will involve multiplying 3 by itself until you figure out that $3^4 = 81$ and so therefore $\log_3 (81) = 4$.

This process of guessing what (whole number) power of a particular (whole) number in order to figure out a logarithm is probably what she is thinking when she refers to "figuring it out by hand". It has served her very well up until now.

Moreover, almost all calculators don't even have a function to calculate general logarithms (though they do have an option to calculate any exponential) so using a calculator to simply find the answer directly is simply not an option when logs are first learned. She just has no experience calculating logs with a calculator and probably doesn't even realise it's possible.

Finally, I daresay she has pretty much no experience calculating logs for anything other than whole numbers. A sudden shift to a log with base an irrational number calculated upon non-whole numbers is probably causing a considerable block.

A possible way to help her through it

I suggest giving her experiences with the aspects of ordinary logs she hasn't had experience with yet.

Firstly, I would talk with her about logs to non-whole number bases. Ask her to calculate things like $\log_{1.5}(3.375)$ and $\log_{0.6}(0.36)$ using her "by hand" method. Then she can come to see that it's ok to use non-whole numbers in the base.

Secondly, I would ask her about calculating non-whole number powers of numbers. I'd ask her how to calculate $2^{1.5}$. You can talk about how $2^{\frac12} = \sqrt{2}$ so it's also $2^1\times 2^{0.5} = 2 \times \sqrt{2}$. Then you can ask her about $2^{1.6}$. As a fraction this is $2^{1+6/10}$ so it is equal to $2^1\times 2^{6/10} = 2\times \sqrt[10]{2^6}$. So now you can calculate any power of 2. Still, you need a calculator to do the roots, of course, which will probably convince her it's ok to do fractional powers using the calculator.

Next, I'd talk with her about how she might go about calculating $\log_2(10)$. By comparing to 8 and 16, you can figure out that the answer is somewhere between 3 and 4. Then the problem is to calculate exactly how far between 3 and 4. Referring back to your fractional powers of 2, $2^{3.5}\approx 11.314$ and so you know that $\log_2(10)$ is below 3.5. You can make it more accurate by trying various values in the power.

At this stage I would get a program that does do general logs (like Microsoft Excel's =LOG(number,base)) just to show that it can be done using some computer programs.

Nearing the end, you can talk about how ln is log base e, and try that out with the computer programs by entering both ln and log base e of various numbers. You can also write a table of values of $e^x$ for various values of $x$, including some decimal values so you can know how big $\ln$ of some numbers will be.

Finally you can show how any logarithm can be calculated from $\ln$ using the change of base law -- students are often very impressed with that one and it ties it all back to their own calculator and the original meaning of log because now they can use their calculator to check answers they did "by hand".

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  • $\begingroup$ I'm glad to read this, because this sound like almost exactly what I did, and it seems to be working out! She has made extraordinary progress in just one day. $\endgroup$ – Bassinator May 18 '15 at 1:23
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    $\begingroup$ I think the lack of a teacher who understands the math well enough herself is likely a significant hindrance... When the teacher dodges the tough questions, it makes it hard to learn. $\endgroup$ – Bassinator May 18 '15 at 1:25
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I think your title reflects your frustration. To me, it sounds like you'd like to show her the mechanics (calculator), but she wants to understand the underlying concepts. I agree with her.

Also, the definition you used for ln x is one of multiple possible definitions. I would use a different definition. ln x means loge x. Definition of logarithm: y = bx is equivalent to logb y = x. This definition is much more appropriate for her level. (For you, the question remains: How can there be multiple definitions? And, how do we prove that the two definitions are equivalent? This is not something she's ready to consider.)

Doing things by hand is an excellent route to understanding. But it is better done with logs using simpler bases - 2, 3, 10, for example. Once she's got logs in general down, she can discuss the meaning of logs base e.

She can start to get a feel for why e is important by analyzing continuously compounded interest.

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    $\begingroup$ (Though, in your proposed definition, there is no explanation of what $e$ is...) $\endgroup$ – Benjamin Dickman May 17 '15 at 17:50
  • $\begingroup$ I agree that the underlying concepts are important. Having worked with her today some more, I think we are heading in the right direction. I have discovered that part of her reluctance to use a calculator (besides wanting to understand where these things come from) is that she had no clue how to use an n-spire. (Heck, I'm majoring in software engineering next year and I am still not a fan of the unnecessary complexity of simple tasks on an nspire. Having shown how to do some of the things on there, and what it is doing behind the scenes a little, we are on the right track now I think. $\endgroup$ – Bassinator May 17 '15 at 18:50
  • $\begingroup$ Also, I don't remember if I wrote it in my question, but I used the explanation of ln(x) = $log_e(x)$. She understands what a log is. $\endgroup$ – Bassinator May 17 '15 at 18:51
  • $\begingroup$ Also, do you really propose teaching someone infinite series (a calculus 2 topic) just to understand how to do ln by hand? As opposed to just explaining what ln is, and where it comes from, and what it gives you, and telling them to trust the calculator to do the number crunching as long as you understand it? I see no more value in that than I do in the value of teaching someone to derive $\pi$ by hand for geometry. Sure, there is knowledge to be found, but at this level, the K.I.S.S. principle applies. $\endgroup$ – Bassinator May 17 '15 at 18:58
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    $\begingroup$ +1 for the base 2 note. When I have a log-confused student, I ask her to hit the reset button, and explain logs with base 2, then jump to base 10, and finally explain the base is arbitrary. We can use any base, but e and 10 are most popular. $\endgroup$ – JoeTaxpayer May 18 '15 at 0:49
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"Nobody ever did them by hand": Presumably Napier did; I think some other people did too. But now about what to tell this student: I think the first thing is to deal with some easier bases and make sure she understands logarithms as inverses of exponential functions. For example, knowing that $2^{10}$ is just a little more than 1000, does she easily see that $\log_{10}2$ is just a little more than $0.3$ (without punching calculator buttons to get $0.301\dots$)? It might be good to do some non-trivial hand calculation of logs to base 10 or base 2, say by bisection, to give her an idea of how it can be done and also how tedious it gets. Next, I'd suggest tackling the thorny problem of defining or explaining $e$. My immediate inclination is to draw the graphs of $x\mapsto a^x$ for several values of $a$, observe their slopes at $x=0$, and introduce $e$ as the value of $a$ that makes this slope $1$. Point out that calculuating $e$ from this definition is tedious at best, but there are alternatives ($\sum(1/n!)$) that work better and are provably equivalent. Once the general notion of logarithm and the specific number $e$ are available, they need to be combined to get $\ln$, which leads to even messier computation. I would not encourage her to do such really messy computations, but I would encourage her to imagine what's involved.

I like this student's attitude of "I want to understand", and I would try to avoid suppressing it, even if the present context is not particularly friendly to this attitude. I would probably give her the explanations she wants wherever possible. Where it's impossible (or infeasible, given time constraints), I would at least indicate why it's difficult and apologize for not being able to give her complete answers.

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    $\begingroup$ "Nobody ever did them by hand, except for the first time." $\endgroup$ – Bassinator May 17 '15 at 23:10
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    $\begingroup$ I do them in my head all the time. For percentage calculations, just remember that log base 1.01 (2) is about 70. $\endgroup$ – gnasher729 May 19 '15 at 8:58
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She is very frustrated because she feels like she needs to know what ln is and how to find it by hand and how to do everything by hand.

I think this is a fantastic attitude, and I suspect it was shared by many great classical mathematicians! Euler, for example, routinely explored new functions, including the logarithm, by doing hand calculations that seem preposterous to us digital natives. When he came up with the power series for the exponential function, one of the first things he did with it was compute $e$ to twenty-three decimal places. Doing calculations by hand often forces you to think about familiar objects in new ways, and it can be a great way to get a feel for how mathematical objects behave.


As an example, think about how you might compute $\log_2 3$ by hand. Observing that $$2^1 < 3 < 2^2,$$ you can sandwich $\log_2 3$ between the integers 1 and 2. For a better estimate—say, a sandwich between sixteenths of an integer—you could try to sandwich $16 \log_2 3$ between integers. Playing around with $16 \log_2 3$, using what you know about how logarithms work, you might eventually notice that it's equal to $\log_2 3^{16}$. If you're willing to use a calculator for multiplication, it only takes a few minutes to find out that $$2^{25} < 3^{16} < 2^{26},$$ sandwiching $16 \log_2 3$ between $25$ and $26$. This shows that $\log_2 3$ is between $\frac{25}{16}$ and $\frac{26}{16}$, pinning it down to within 2%.


One of the ways people first ran into the natural logarithm was by thinking about slow exponential growth, which can be done very computationally. Let's say you're watching an algae bloom that grows at a rate proportional to its size, increasing its mass by $\frac{1}{10}$ of its current mass every day. In symbols, the size of the bloom after $p$ days is $$\left(1 + \frac{1}{10}\right)^p.$$ Using a calculator for multiplication, it's not hard to compute that the bloom will double in size every 7 or 8 days.

What if the algae reproduce more slowly? Let's say the bloom only grows by $\frac{1}{20}$ of its current mass each day. More calculator work reveals that the doubling time is 14 to 15 days.

If the bloom grows by $\frac{1}{30}$ of its mass each day, the doubling time is 21 to 22 days. A growth rate of $\frac{1}{40}$ gives a doubling time of 28 to 29 days. A growth rate of $\frac{1}{n}$ always seems to give a doubling time close to a constant multiple of $n$! That constant multiple, which from our calculations appears to be around 0.7, is the natural logarithm of two.


We've found a number $\ln 2$ with the property that it takes about $n \ln 2$ days for something growing at a rate of $\frac{1}{n}$ per day to grow by a factor of 2. In other words, $$\left(1 + \frac{1}{n}\right)^{n \ln 2} \approx 2,$$ an approximation that gets better and better as $n$ grows. This phenomenon isn't limited to doubling times. For any growth factor $x$, there's a number $\ln x$ with the property that $$\left(1 + \frac{1}{n}\right)^{n \ln x} \approx x$$ more and more accurately as $n$ grows. Looking at this expression, you might suspect that $$\left(1 + \frac{1}{n}\right)^n$$ approaches some constant as $n$ grows. For some values of $n$, you can evaluate the expression above very quickly using clever tricks. For example, you can evaluate it for $n = 256$ by squaring $1 + \frac{1}{256}$ eight times. If you do this carefully (a calculator with a squaring button helps a lot), you can see that as $n$ grows, the expression does seem to approach a constant—something around 2.71.


With these examples, I hope I've managed to convince you that hand computations are a fun way to get a feel for the natural logarithm. The same goes for lots of other functions you'll encounter in your classes. If you ask your classmate to keep challenging you to understand the computational fundamentals of the new functions you encounter, I think you'll learn a lot!

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What about how Euler did it to find logarithms by hand. Though you may consult the paper cited personally, I'd like to add a few comments. First, notice that how Euler reduce the problem of finding logarithms to the "easier" problem of finding square roots using repeatedly $\log \sqrt{xy}=\frac {\log x + \log y}{2}$. Moreover, it seems funny (from a modern viewpoint) how he performs the process until he runs out letters! However, it also shows that how even such a seemingly simple thing as indexing letters has been developed during the years until it has reached the point we know it. Of course, the notion of logarithm has a more complex history, an important part of which involves calculating by hand. Respecting this part is respecting the mathematicians who helped us to understand logarithm as we know it today. Moreover, It encourages our students not to take for granted what they have (i.e., calculators). It can be also used as a bridge to understanding how calculators calculate logarithms. To be honest, I am so fond of the "handy" history of logarithms that whenever I teach it I ask students to do it at least once as Euler did it. I also give one minute silence to respect his work, Napier's and Brigg's.

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How about the following characterization of $\ln$:

$\ln: \mathbb{R}^+ \to \mathbb{R}$ is the unique continuous function satisfying the following properties:

  1. $\ln(xy) = \ln(x)+\ln(y)$ for all $x,y \geq 0$

  2. $\ln(1+\epsilon) \approx \epsilon$

Property $2$ is a bit wishy washy (should really be $\frac{d}{dx} \ln(x) \big|_{x=1} = 1$), but will let her compute stuff.

You could prove that $\ln(1) = 0$.

Then prove $\ln(x^n) = n\ln(x)$ for natural numbers $n$.

See if she has an idea for approximating $\ln(2)$. My idea would be to write $\ln(2) = \ln((1+0.01)^n)$ for some very large $n$. Then $\ln(2) = n\ln(1+0.01) \approx n(0.01)$. You could attempt to find this $n$ with an excel spreadsheet, and then confirm with the calculator button. You could obviously do the spreadsheet "by hand", so it is not cheating much.

Do the same thing for $\ln(3)$. Notice that $\ln(2)<1$ and $\ln(3)>1$, so there should be some number whose natural log is 1. This is the definition of $e$ in this framework.

She might notice that she is really making a table of values for $\ln$ when she raises $(1+\epsilon)$ to higher and higher powers. Approximate $e$ to varying degrees of accuracy by decreasing $\epsilon$. Compare to the calculator value of $e$.

This could develop into months of mathematical exploration. How does this definition of $e$ relate to other definitions? How is $\ln$ related to the exponential function? How do we know that a function satisfying these properties even exists? How does this characterization relate to solving exponential equations? How does this tie into the area picture?

You may also be interested in my math.stackexchange post here, which is at a level appropriate for a calculus student https://math.stackexchange.com/questions/498339/demystify-integration-of-int-frac1x-mathrm-dx/498790#498790

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    $\begingroup$ If you want to make property 2 exact, but don't want to introduce derivatives, you can formulate it as "$\ln(1+\epsilon)/\epsilon\to1$ as $\epsilon\to0$", or just as "the slope of the curve $y=\ln x$ at the point where it crosses the $x$ axis is 1." It may also be worth noting explicitly that the first property is shared by $\log_b$ for any base $b$, while the second one is what distinguishes $\ln = \log_e$ from other logarithm bases. $\endgroup$ – Ilmari Karonen May 19 '15 at 13:07
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    $\begingroup$ @IlmariKaronen I think the statement involving limits is perhaps not appropriate at a precalculus level, and the slope statement is just as wishy washy as my statement. I agree that it should be made clear that property 2 is what distinguishes the natural log from all other logarithms. $\endgroup$ – Steven Gubkin May 19 '15 at 13:52
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There's a few things here I want to comment on:

-I think your friend should know what ln means, but I don't think she should care about whether she has to use a calculator. It's important to understand at least a basic definition of ln(x) (even as "e to the power of what number equals x?") because otherwise she'll have difficulty solving logarithmic equations (or she'll only be able to solve them by repeating memorized procedures).

-The idea that decimal expressions are somehow preferable is a huge misconception (usually reinforced in mathematics teaching). ln(5.4)/8 is a perfectly acceptable answer in some contexts, ~0.21 is an acceptable answer in others. They both provide useful information about that number - for example, the approximation is useful for situating it on the number line (so you get an idea of how big it is; ie, you know ln(5.4)/8 is less than 1 but greater than 0). But it can never be "exact" because ln(5.4)/8 has a non-periodic decimal expansion.

-She wants to be able to "do it herself", without the assistance of a machine. But a calculator doesn't magically know the decimal approximation for ln(5.4)/8. It uses an algorithm to find it, the derivation of which has to be well beyond the scope of the course she's taking. Surely she realizes that there is other, way more complicated mathematics that she has yet to be exposed to and couldn't possibly be exposed to (either because she's unable to grasp it yet or because there's nowhere near enough time to teach that much material that early on).

-I like your definition of ln(a), and if explained properly I think it could be helpful in this situation (I'm making no judgments about whether you explained it properly, whether she just didn't "get it", etc). You define it as the definite integral of 1/x from 1 to a, which is ln(a)-ln(1)=ln(a). So if she wanted to find an approximation for ln(5.4), she would just end up back at the same spot where she'd have to plug ln(5.4) into her calculator. On the other hand, she could approximate it "on her own" by drawing a bunch of suitably thin rectangles and summing their areas (to put it informally) which anyone can do, knowing the area of a rectangle, the function y=1/x and knowing how to plug numbers into a function. This helps demonstrate the futility of approximating it without calculator assistance, and why understanding where these values come from is unnecessary (ie, assuming she figures out how to do all this, she isn't going to feel more fulfilled or more able to solve problems involving logs).

I'm sure this could have been more consise, so sorry about that.

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Have her use tables for a set of exercises. It is sort of a halfway solution. You don't have the total black box of the calculator (see numbers next to each other in the book). Also, using the tables requires you to do that whole mantissa and characteristic. You get a bit more feel for the things.

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