Category Archives: In Depth

Logs and the Change of Base Formula

Prior to calculators, the standard way for determining a logarithm precisely was to use a look-up table for values. However, such tables were usually only printed for the common most bases, particularly 10 (log). To determine a logarithm for a different base, you needed to first convert to that base.

Luckily, it’s very simple to do so. The change of base formula is: \[\log_a b = \frac{\log_c b}{\log_c a}\]

This works for any base, but it’s most typical to use it for 10 and e. For instance, if you want to find \(\log_6 29\) using a log (base 10) table, you find \(\log 6\) and \(\log 29\), then divide. That is, \[\log_6 29 = \frac{\log 29}{\log 6} \approx \frac{1.4624}{.7782} \approx 1.879\]

Side note: You’ll notice that the log table I linked only goes up to 9.99. So how did I use it to find \(\log 29\)? I looked up \(\log 2.9\) and then added 1. This is why common log tables were most common for this purpose: They’re more flexible than natural log tables. You only need the values from 1.000 to 9.999 to find the log of any positive real number.

Since many calculators only have log and ln buttons, it’s useful to know this formula. Even for the calculator we use in class, you might find that it’s easier to use this technique instead of using [2nd][Window/F1][5].

You’ll get the same answer either way because the calculator is performing a change of base anyway. Using [2nd][Window/F1][5], find \(\log_0 0\) and \(\log_1 1\). You’ll get an error for each, but it’s a different error: In the first case, the error is Error: Domain because 0 is not a valid base. In the second case, the error is Error: Divide by 0. If the calculator were doing the logarithm directly, it should give another domain error (because 1 is not a valid base). By the change of base formula, though, you would calculate \(\log 1 \div \log 1 = 0 \div 0\).

Proof

The change of base formula seems weird: Why does it work? How does making the base into its own log work? This is magic!

It’s not magic, it’s math. Logarithms do sometimes work in ways we might not immediately predict. The proof is remarkably short, given how unusual it looks.

First, let’s replace each part of the formula with a variable. That is, we’ll let \[ x = \log_a b \\ y = \log_c b \\ z = \log_c a\]

Let’s rewrite each of these into exponential form: \[ a^x = b \\ c^y = b \\ c^z = a\]

Replacing \(a\) in the first line with \(c^z\) from the third line, then using the transitive property on the first two lines, gives us: \[ (c^z)^x = c^y \]

Because of the properties of exponents, we can rewrite \((c^z)^x = c^{xz}\). Two exponential expressions with the same base (other than 1 and 0, which are excluded by definition) have to have the same exponent, that is: \[ xz = y \Rightarrow x = \frac{y}{z} \]

We can now replace the variables as we defined them in the first step, giving us: \[\log_a b = \frac{\log_c b}{\log_c a}\]

which is the change of base formula.

Supermutations

A supermutation is a number that contains all the possible permutations of a set of digits. For instance, 123121321 contains all the possible permutations of 1, 2, and 3 (123, 132, 213, 231, 312, and 321). This video is an interesting exploration of permutations, beyond what we’re doing in class.

Synthetic Division with a Quadratic Divisor

In class and in the book, it is said that synthetic division only works if the divisor is a linear function, i.e., something that can be written in the form (x + k). That’s not true, but synthetic division with higher-order polynomials is a little more complicated.

Here’s an example of synthetic division using a linear divisor. Consider \((3x^3 + 2x^2 – x + 7)\div (x – 2)\). Here it is using synthetic division: \[\begin{array}{cccccc}2&|&3&2&-1&7\\&|&\downarrow&6&16&30\\&&——&——&——&——\\&&3&8&15&37\end{array}\]

The quotient is \(3x^2 + 8 + 15 + \frac{37}{x – 2}\).

Synthetic division for higher order divisors relies on the same concept, but needs a new line for each coefficient. For instance, a quadratic divisor uses two lines instead of one; a cubic divisor uses three lines; and so on.

For instance, \((x^2 + 3x – 7)(x^2 – 5x + 2) = x^4 – 2x^3 -20x^2 + 41x – 14\). To get a remainder, we’ll change the last two terms. To the left, we’ll reverse the signs of both of the coefficients of the lower terms of the divisor; as before, we’ll write all of the coefficients of the dividend: \[\begin{array}{cccccccc}-3&7&|&1&-2&-20&40&-10\\ & &|&\downarrow&-3&15&-6& \\ & &|&\downarrow&&7&-35&14\\ & & &——&——&——&——&——\\ & & &1&-5&2&-1&4\end{array}\]
In the second row, we use opposite of the coefficient from the \(x\) term (that is, -3) as a multiplier; in the third row, we use opposite of the constant (that is, 7).

This gives us a result of \(x^2 – 5x + 2 + \frac{-x + 4}{x^2 + 3x – 7}\), which is the same thing we get from long division.

Notice that, regardless, the lead coefficient (\(a\)) of the divisor must be 1. You can use synthetic division with, say, a divisor of \(4x – 5\), but you’d have to use \(x – 5/4\) instead, and then divide each coefficient of the result by 4.

For example, consider \((2x^3 + x^2 – 17x + 14)/(2x + 7)\). This is what synthetic division yields: \[\begin{array}{cccccc}-7/2&|&2&1&-17&14\\&|&\downarrow&-7&21&-14\\&&——&——&——&——\\&&2&-6&4&0\end{array}\]

The actual quotient is \(x^2 – 3x + 2\), with no remainder, which is what we get from dividing each of the resulting values by 2.

Other Cultures and Math

I’ve mentioned in class that German uses slightly different symbols than we do. Here’s a screen shot from a German video on linear functions.

Most of this is recognizable. The 1s are different than ours, but there are two bigger differences:

  1. The original problem is: Given the point \(P(1, 2)\) and the slope \(m = -2\), what is the graph? In this case, the point is given as \(P(1|2)\).
  2. In the last step of solving for b, instead of writing \(+2\) on both sides, the instructor wrote \(| +2\) to the right of the entire equation.

I find it interesting to look at how other countries and cultures represent mathematics. The notation is indeed a language, and like any language there are variations around the world. The underlying concepts are universal, though.

Graphing calculators for your smartphone

Here are three options for free graphing calculators on your tablet or smart phone. If none of these are to your liking, just search the store for “graphing calculators”. The information below is based on the Android versions running on my tablet.

  1. Desmos (available for iOS and Android, as well as in browsers)
    • Basic functions are available on the main screen.
    • To get the third root, press “functions” -> “misc” and pick the nth root button. Then use 3 for n.
  2. GeoGebra (available for iOS and Android, as well as in browsers)
    • Basic functions are available on the main screen.
    • To get the third root, press “f(x)” and pick the nth root button. Then use 3 for the first blank.
  3. Wabbitemu (available for Android, as well as computer installation)
    • This allows you to load a calculator that works just like our class calculators! I would recommend this for tablets, but it’s probably too small for a phone.
    • Once you’ve installed and run the app, select “Help me create a ROM using open source software” and click “Next”.
    • Select “TI-84 Plus C SE” and click “Next”.
    • You can now touch buttons just like the class calculators.

 

Vi Hart on Logarithms

In this video, Vi Hart talks about the multiplication scale, which gives us logarithms. She speaks quickly and poetically; don’t expect to understand all of it at this point. If you just want to hear the part about logarithms (which applies to the slide rule that I was discussing today), jump to the 5:00 mark.

Simplifying Radicals on the TI-84 CE

Unfortunately, there doesn’t seem to be a pre-existing function on the TI-84 CE to present a simplified radical. You can write a program, which I provide here, but this takes a lot of work to simply enter into the calculator that I don’t advise it. However, I’m presenting it here to give you an idea of how to program the calculator, in case you’re interested.

To create a program of your own, including entering this one, press the prgm button, then select NEW and 1:Create New. Give it a name (I called this SIMPRAD for “Simplify Radical”).

Then enter the code below, not including the line-initial colons. These colons represent the start of a new line.

:Input "RADICAND? ",D
:1→C
:"+"→Str3
:If D<0
:Then
:-D→D
:"i"→Str3
:End
:If D>0 and fPart(D)=0
:Then
:While fPart(D/4)=0
:C*2→C
:D/4→D
:End
:For(E,3,√(D),2)
:While fPart(D/E^2)=0
:C*E→C
:D/E^2→D
:End
:End
:"?
:For(A,1,1+log(C))
:sub("0123456789",ipart(10*fPart(C*10^(-A)))+1,1)+Ans
:End
:sub(Ans,1,length(Ans)-1→Str1
:"?
:For(A,1,1+log(D))
:sub("0123456789",ipart(10*fPart(D*10^(-A)))+1,1)+Ans
:End
:sub(Ans,1,length(Ans)-1→Str2
:If Str3="i"
:Str1+Str3→Str1
:If D>1
:Str1+"√("+Str2+")"→Str1
:Disp Str1
:Else
:If D=0
:Then
:Disp "0"
:Else
:Disp "INVALID"
:End
:End

I won’t go through all the entry details; some of the characters can be entered from keys on the calculator, while others require going to specific menus. If you really do want to enter this into your calculator, search around or ask me for specific items.

Let’s look at how each section of this code works.

:Input "RADICAND? ",D
:1→C
:"+"→Str3

The lines above ask the user for the number to be simplified. For instance, if you want to simplify \(\sqrt{412}\), you would enter 412. When the program is done, D will hold the radicand and C will hold the coefficient. Str3 will let us know if the initial radicand is negative.

:If D<0
:Then
:-D→D
:"i"→Str3
:End

The lines above allow for imaginary roots.

:If D>0 and fPart(D)=0
:Then

We will only process positive integers this way; 0 and non-integers will be handled separately.

:While fPart(D/4)=0
:C*2→C
:D/4→D
:End

There are two approaches we could use: Have a list of primes that we walk through, or test 2 and then all odd integers greater than 1. For ease of programming, I’ll do the latter. So these lines divide the radicand by \(2^2 = 4\) until doing so results in a non-integer.

:For(E,3,√(D),2)
:While fPart(D/E^2)=0
:C*E→C
:D/E^2→D
:End
:End

These lines divide the radicand by \(3^2 = 9\), \(5^2 = 25\), and so on up to the square root of the radicand, moving on to each new odd number when dividing results in a non-integer.

At this point, we have what we need. If we were willing to have ugly output, we could pretty much stop here. Most of the rest of the code is to make the output attractive. Because the TI-84 CE couldn’t easily convert a number into a string (characters on a screen that aren’t numbers), and couldn’t connect a number to a string, we have to do this. A recent OS update changed this, but I’m providing code that works for all the calculators we have in the room.

:"?
:For(A,1,1+log(C))
:sub("0123456789",ipart(10*fPart(C*10^(-A)))+1,1)+Ans
:End
:sub(Ans,1,length(Ans)-1→Str1

The lines above convert the coeefficient from the number C into the string Str1.

:"?
:For(A,1,1+log(D))
:sub("0123456789",ipart(10*fPart(D*10^(-A)))+1,1)+Ans
:End
:sub(Ans,1,length(Ans)-1→Str2

The lines avove convert the radicand into Str2.

:If Str3="i"
:Str1+Str3→Str1
:If D>1
:Str1+"√("+Str2+")"→Str1
:Disp Str1

The lines above create a string like 4i√3. The rest of the code handles 0 (in which case, just display 0) and non-integers (in which case, display “INVALID”).

:Else
:If D=0
:Then
:Disp "0"
:Else
:Disp "INVALID"
:End
:End

I think this gives an interesting overview to programming the TI 84. If you have a personal calculator and want to store this, feel free.