How should I teach logarithms to high school students?

Question

I have some very basic questions about how to teach the logarithm to high school students:

First of all, is it better to introduce it as a function with a graph or is it better to treat it like a black box with certain properties at first with which to solve exponential equations?

And once the best starting point has been used, how can I lead my students to a deeper understanding of logarithms? Are there any particularly conducive methods? Like e.g. have them solve twenty problems for each rule of logarithm?

One of my problems with logarithms is that after I tell them that the logarithm is the inverse function of the exponential then there is nothing more to think about: If they are supposed to solve an equation then they can just apply the logarithm to both sides of the equation. That's boring.

Ideally, I'd like to pose interesting problems only.

Answer 1

Coming from the perspective of someone who reteaches this material at the college level, neither the graph perspective nor the list of properties perspective really translate into a deep understanding later. For the former, all you get is an understanding that the graph goes up, but not a lot, and is too vague to really be useful. For the latter, they don't remember the rules and the "big picture" that the logarithm transforms multiplication into addition (i.e. is a group homomorphism from $(\mathbb{R}^+,\times)$ to $(\mathbb{R},+)$) gets spotty understanding.

So, I'd like to introduce another interpretation for the logarithm, analogous to an interpretation for division. That is, you can interpret the integer part of $a / b$ as the number of times that you can subtract $b$ from $a$ and stay above $0$. Similarly, the integer part of $\log_b(a)$ is the number of times that you can divide $a$ by $b$ and stay above $1$. There are several things you can get from that which might aid understanding:

• $1+\lfloor \log_{10}(n) \rfloor $ is the number of decimal digits of an integer $n$.
• "20" questions: If you have $n$ things and want to pick out individuals by asking yes/no questions, $\lceil \log_{2}(n) \rceil$ is the minimum number you have to ask to distinguish every one of them. You can tie this into questions about how much personal information is required to distinguish every person in a country.

Answer 2

How do you teach students about the operator $\sqrt[3]{}$? It's a similar operator in many ways; when dealing with cube roots I try to show them these things:

1. $\sqrt[3]{x}$ asks "which base to the third power gives us $x$?"
2. $f(x) = \sqrt[3]{x}$ is a one-to-one function, and $y=f(x)$ has a graph of a certain shape.
3. $\sqrt[3]{}$ is the inverse operation to the cube, so it's useful for undoing cubes.
4. $\sqrt[3]{}$ does not distribute over addition, but it does over multiplication.

So, for logarithms I do the corresponding things:

1. $\log_3{x}$ asks "which exponent with base 3 gives us $x$?"
2. $f(x) = \log_3(x)$ is a one-to-one function, and $y=f(x)$ has a graph of a certain shape.
3. $\log_3$ is the inverse operation to exponentiation with base $3$, so it's useful for undoing exponentiation with base $3$.
4. $\log_3$ does not distribute over addition, and whoa, look at what it does with multiplication.

A good exercise following this presentation would be:

Consider these two equations: $x^3 = 729$; $3^x = 729$. How do they look similar? How are they different? Solve each of them.

Answer 3

I would start with explaining exponentials as repeated multiplication. For example, we look at the sequence $2^1=2$, $2^2=2\times 2$, $2^3=2\times 2\times 2$... and call it one 'two', two 'twos', three 'twos'. Then show the rule for multiplication goes like $(2\times 2)\times (2\times 2\times 2)$ and say that we find the total number of twos by adding up the number of twos in each bracket. $2^2\times 2^3=2^5$. So we can do multiplication of blocks of twos by adding the number of twos in each block. We have turned multiplying into adding. And then the 'logarithm to base 2' is the number of twos in a block.

Repeat for several different numbers, like powers of 3, 4, 10 to establish the concept for whole numbers. 10 should already be familiar to them, so it links it back to existing knowledge.

Then we 'fill in the gaps' between the whole numbers. We show that square roots act like they are half way between the whole numbers. $2^{1/2}=\sqrt{2}$, $2^1=\sqrt{2}\times \sqrt{2}$, $2^{3/2}=\sqrt{2}\times \sqrt{2}\times \sqrt{2}$, $2^2=\sqrt{2}\times \sqrt{2}\times \sqrt{2}\times \sqrt{2}$. The logarithms go up in steps of a half. And if we plot the points on a graph, we can gradually fill in the gaps to get a smooth curve.

Then for any number, we can use the smooth curve to describe it as being very close to some number of twos (possibly fractional) multiplied together, and use the previous trick to turn any multiplication into addition. Slide rules are a very good example to demonstrate how the knowledge can be applied. Students can make a simple one themselves, and you can show them a professional one and maybe talk a little about the history of calculation before calculators. And for something more up to date, compound interest on loans is very important to anyone who wants to buy something expensive like a house. '3% per annum compound for 20 years' means a block of twenty 1.03's multiplied together. If you want to borrow \$1000 now and pay back no more than \$1100 in six months time, what's the maximum annual interest rate you can afford? Gambling is another fun example - how many times do you have to toss a coin before the odds of winning every time drop below one in a million?

As a rule, any new concept should be first introduced with concrete examples, then with pictures or visual metaphors, it should ideally be connected to a real world problem they can relate to, and finally turned into an abstract definition. Revisit the whole concrete-pictorial-abstract build-up several times over a period of a few days, with gaps in between. People remember things they find they repeatedly need to know - the act of recalling it affixes it more firmly as a long-term memory.

Answer 4

Instant intuition giver: Logarithm computes the number of digits.

One could start by letting the students think how such a function should behave.

One possible idea: That's just for whole numbers. Is there a way to interpolate? (Turns out yes, there is -> give the formal definition for decadic logarithm.)

Another one: Different number systems. Binary numbers are longer than their standard power-of-ten equivalent, shouldn't the logarithm be different. (Answer: depends on the number system but it's only a constant factor, generalize to base-N logarithm.)

Once these two questions have been answered, the full definition is there and all the cool properties like "is the inverse of exponentiation" and that it lowers multiplication to addition can be introduced.
The students can then use the number-of-digits intuition to validate that all these logarithm laws do, in fact, make sense to them.

Caveat: This would have been the best approach for my own personal intuition, which may or may not align with that of others.
However, I do know that starting with the raw definition "it's the inverse of exponentiation" is very hard to grasp in high school, I was not the only one who was struggling with it. I suspect that's because in high school, the only examples of invertible functions are addition and multiplication, where the inverse function can be conceptually split into two operations, "take the inverse of the second operand" and "apply the original operation using that inverse"; this tactics will fail for logarithms, so people will be left with no way to build an intuition.
That number-of-digits-plus-sensible-interpolation intuition has been carrying me through decades, so I am mildly confident maybe it is useful to other students as well.

Answer 5

All of the math explanations here are very good. And they all use math to explain math. Sometimes it is helpful to understand the rationale; concept or the "Why was this invented / Why does someone need this;" or even the 'real world' explanation behand the math.

I had a story I used to tell. It went something like this:
Suppose you need to count to a very large number. Let's say [sic] "A Jillion."
You start out counting 1, 2, 3, 4, etc. up to 10.
Now, the ones digit repeats: 11, 12, 13, and so on. But this is now the 'uninteresting' part. We know this already. So, this time, you count by Tens: 10, 20, 30, 40, etc. We are not truly ignoring the numbers in between, we know they are there, but they don't necessarily add anything to the count.
When we got to 100, we again change our scale and count by hundreds: 100, 200, 300, 400, etc.
At 1,000, we re-scale and count by thousands: 1,000; 2,000; 3,000; 4,000 etc.
We do this until we reach "A Jillion."

This is the concept of Base 10 Logarithms.

Why do this?
If I am poor, and someone asks me how much money I have, I will tell them in dollars and cents: \$4.60, because that is the scale I need to describe the number.
If, on the other hand, I am a multi-billionaire, and someone asks me how much money I have, dollars and cents don't really matter: $4.6 Billion, because Billions is the scale I need to neatly describe the number.

Why do I need to do that? It is convenient and efficient, but it is also useful in other areas. Bill Nye has a great explanation using the Richter scale (which is a logarithmic scale.)
Bill Nye the Science Guy - Earthquakes (Richter scale)

So, basically, logarithms allow me to adjust the scale for very large or very small numbers.

Now, introducing "The Math" to work out these scenarios will, hopefully, make more sense.

$0.02

Answer 6

Although I am not a teacher, I would recommend going with the historical perspective. Logarithms were invented by John Napier (see https://en.wikipedia.org/wiki/Logarithm) basically to make multiplication easier. Instead of having to multiply several large numbers, you looked up their logarithms in the table, then only had to add them and consult the table again in reverse direction with the sum to find the product.

Bring a book with the logarithmic tables and demonstrate the process.

This was how I learned about logarithms from my father and also how I taught them to my children. The process seemed utterly magical to me and really stuck in my memory.

Students today will also be amazed at the concept of doing manual calculations and how much work they were but also how important.

Once the properties of logarithms as aid to multiplication have been established, you can start deriving what properties they need to have, e.g. that base 10 logarithms increase by 1 for each factor of 10, etc..

That builds the bridge to the relationship of logarithms and exponentiation, and from there you can introduce logarithms of different bases and finally the natural logarithms.

Answer 7

I think it would be more useful to teach logarithms as the inverse of exponents¹, and compare most of the properties of logarithms with the properties of exponents, because most of the properties of logarithms are related to those of exponents, or those of other logarithmic properties.

Here’s a list of common (as in often talked about, and not that complicated) logarithmic properties and how they relate to properties of exponents.

1. Logarithms are the inverse of exponents. $log_b(b^x)=x$ and $ b^{log_b(x)}=x$ is probably the simplest way to describe this in relation to exponents, and if the students have already been thought inverse functions, they shouldn’t have much trouble with this because they’ll already understand that $f^{-1}(f(x))=x$
2. The fact that $log(a)+log(b)=log(ab)$ can be described by comparing it to a similar law for exponents. In exponents, $10^a*10^b=10^{a+b}$ is the related law.
3. The logarithm rule saying $b*log(a)=log(a^b)$ can be compared to the exponent rule $(a^b)^c=a^{bc}$, or alternatively it can be considered as an extension of $log(a)+log(b)=log(ab)$ because multiplication is repeated addition (excluding unusual cases like imaginary numbers, which normally are not thought with basic level logarithms).
4. The logarithm rule saying $log(a)-log(b)=log(a/b)$ is related to the exponent rule $\frac{10^a}{10^b}=10^{a-b}$
5. The rules $log_x(x)=1$ is an extension of the rule that $log_x(x^b)=b$ and b is just equal to one.
6. The rule that $log_m(1)=0$ is related to the exponent rule that $m^0=1$
7. The logarithm rule that $\frac{log_a(b)}{log_a(c)}=log_c(b)$ cannot be compared to any exponential rule that I know of, so you can either make your students simply memorize it, or if you don’t plan on your students getting tested on this rule (since from what I can tell, it isn’t as popular as the other rules), just not teach it to them.

Important note: this idea falls apart if you haven’t thought your students all of the exponent rules and what inverse functions are.

¹I think this because this NBC article says:

We remember things because they either stand out, they relate to and can easily be integrated in our existing knowledge base, or it’s something we retrieve, recount or use repeatedly over time

(Emphasis mine)

So relating logarithms to exponents will make them easier to remember.

Answer 8

Slide rules are a good tactile method of exploring logs. You can use them to calculate the $log_ba$ and $b^a$, and $a\cdot b$. They also visually illustrate the properties of logs, such as $\log(xy)=\log(x)+\log(y), \log(x^y)=y\log(x), \log_yx=\frac{\log_bx}{\log_by}$.

Answer 9

Log-linear and log-log graph paper can help visualization. (If paper is passé, you can find templates on the web.) Plot the same function on the different scales.

Examples help. Familiar examples.

E.g., the signal strength bars on mobile phones are decibels and are logarithmic.

E.g., the Richter scale. It may not be so familiar unless you live in California, but cool, and in newspapers.

Answer 10

This is a pretty broad question, but here are some suggestions.

Start with some remediation of the notion of a proportionality. In the K-12 curriculum, most students only see direct proportionalities, not proportionalities of the form $y \propto x^n$ with $n\ne 1$. Do an example like a comparison of the number of jelly beans that can fit in one jar compared to another, when the second jar has the same shape but double the linear dimensions. If they don't understand this sort of thing, then it's basically a crippling issue when they get to logarithms. Convince them to reason about things like $y_1/y_2$ and $x_1/x_2$, using an example where using these ratios makes it easy to solve the problem in your head, while not using ratios makes it a mess. E.g., for the jelly bean problem, show how awful it gets if they insist on explicitly writing down the formula for the volume of a cylinder and then just cranking out the algebra. Point out that ratios are unitless, which is a big win. Point out that the factors like pi are destined to cancel out. This sort of thing is necessary, or else they will never talk or think about ratios.

Give them tasks in which they translate the identities into the simplest and most transparent possible example using arithmetic. For example, $\log(a^b)=b\log a$ can be exemplified using $10^3$ and log base 10.

Connect to STEM examples such as pH, radioactive half-life, or the number of bits needed to represent an address on a computer with $n$ bytes of memory. Another computer-sciency example is the number of steps needed in order to look up a word in the dictionary, if you follow an algorithm where you repeatedly bisect.

Do the technique where you extract an unknown exponent from a log-log plot of data points. This is most transparent if you start with artificial data and use log base 10, e.g., the data points are (1,5), (10,500), (100,50000). After showing a simple example like this, show an interesting real one. For example, people have measured the amount of effort required for a team of computer programmers to write n lines of code.

Print out a simplified slide rule and hand out scissors, have them use it to do multiplication.

Pose this question to them. Hey, suppose your friend says, "I learned how to manipulate logs in high school, but I never understood what it was about." Explain to your friend in 10 words or less. My own answer would be something like "Logs turn multiplication into addition."