Q: My students find it difficult to remember how to switch between exponentials and logarithmic equations and vice versa. That is, when writing x= ay and then writing as y = loga(x). Even worse is trying to remember the formula for changing bases in logarithms. They seem to need to get the formula out and substitute in the values, any suggestions?
A: Logarithms have their origin in simplifying complicated calculations (Greek "logos" means "to calculate" and "arithmos" means "a number". A logarithm is "a calculating number").
There isn't a need to remember a rule for change of bases as long as you have a basic understanding of how logarithms work. Let us begin with the first part of your question. Quite often students get confused in the use of the language: confusing the meanings of base, logarithm and exponent. If we talk of the power as the index number then we have a way of remembering exponent, as both words contain an "x": indeX and eXponent. The base is the number that is raised to a power; that is, the number that has to be multiplied by itself the amount indicated by the exponent. In the same way, we can think of multiplication being shorthand for repeated addition
5 x 2 = 2 + 2 + 2 + 2 + 2
we can think of exponents being shorthand for repeated multiplication. 25 2 x 2 x 2 x 2 x 2
In this example, 5 is the exponent and 2 the base. Your students might be interested to see a set of log tables.
Where we have an exponential function such for example p = qt, the logarithmic function logqp=t (q
0) is the inverse function. So how to get this inverse function without trying to remember what goes where? I thought this illustration might help.
They need to know that lognn = 1
Let us look at an example: 23 = 8
The base is 2 so we log2 both sides: log2(23)= log2 8
We rewrite this as 3log22 = log28 (from the rule loga(xylogax + logay which is log23 = log2 (2 x 2 x 2log22 + log22 + log22 = 3log22) , we also have that log22 = 1 which gives 3 = log28.
Students can practice at www.themathpage.comaPreCalclogarithms.htm
I employ the same method for changing bases, a little longer than straight substitution but it has its basis in understanding the manipulation of logs. The formula for changing bases is
loga b =logkb
assuming that a, b and k are all positive real numbers and that a - 1 and k - 1 and k is any valid base.
A real application for changing base: the Udden-Wentworth (generally called the Wentworth scale) is used by geologists to classify sediments and sedimentary rocks from clay to boulders. The scale is in millimetre factors of 2. So for example in a sample largely made up of stones measuring from 4 to 16 mm (22 to 24 mm) diameter would be classified as "pebbles". The scale uses f as a unit of measure. Calculated as f = - log2 (grain diameter in millimetres).
Let's say we have a stone that is 9mm in diameter and we want to find out how many f units this is. We have f = - log2(9). Using the laws of logs we have f = log2(9-1), giving f = log2 19. Take the exponent of 2 of both sides 2f = 2 log2 19. Leaving 2f = 19.
Now taking the log of both sides we have log102f = log1019 . And using the index law of logs,
f log102 = log1019
f = log1019
Now this can be entered onto the calculator to give an approximate of value -3.17. This is - 3.17f units, which is classified as pebbles.
To find out more about the Wentworth scale in geology, visit www.eos.ubc.cacourseseosc221sedsilisiligsize.html
Recently a headteacher told me how difficult he had found it to fill a maths post despite his being a great school with good results and a strong maths department in a lovely area of south-west England. There was only one applicant for the first interview, then one turned down the post and there were only four for the final set of interviews. This had taken months. This made me reflect on the crisis which faces us and adds to the pressures in the classroom. My reflection below was for the child in the school where a vacancy remains unfilled.
The Crisis 2005
Mathematics classes. Passes.
No one to teach us, to reach us, mathematics classes; passes.
Vacancies grow, upward flow,
mathematics classes: passes.
We need to learn, teacher yearn,
mathematics classes, passes.
One applicant, reluctant,
Mathematics classes? Passes.
Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses.
Email your questions to Mathagony Aunt at firstname.lastname@example.org Or write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX