# Using only pencil and paper work out  984 876 543 210 x 875 622 375 933. You have 2 minutes . . . go! Have you worked it out yet: 984 876 543 210 x 875 622 375 933 ? Compare methods in the class - have we all used the same method? Now compare your answers, are they all the same?

Most people tend to make a mistake at some point in that calculation (maybe you did too?) and two minutes isn't enough time (for most people) to complete it! Another solution is required.

### Overview of this Activity

Option1: Mechanical Calculators

There was a time when the German produced Schubert calculator was high-tech (late 1950s early 1960s). I have one of these mechanical calculators in my classroom. Challenge: can you work out/find out, how it works and be ready to show the class next lesson? Option2: Log Tables (turn every number into a power of TEN)

For this you’re going to need your laws of indices/exponents (this activity may help remind you: Prime and Powers Investigation). Can you remember what: 2.54624 x 2.54628  =  ______________ ?

Hopefully you remember your “laws of indices/exponents”? Write a quick summary on a sheet of paper.

### Using Log Tables

We’re going to use a simple example, to illustrate how to use log tables ("zoom in" to enlarge the table below) effectively to do: 2.07 x 2.08.

a) Use the “Log Table” below to find what power of 10, 2.07 and 2.08 are equal to (then see the instructions below for the next steps). http://math.ucsb.edu/~myoshi/logtable.bmp

• Now find the sum of these powers in the table and read off what number this new power corresponds to.
• Is the answer the same as what you get if you do do 2.07 x 2.08 using a long multiplication method? Test it: by typing in 10^the sum of the power of ten for 2.07 and 2.08, into your GDC.

b) Try using the log tables again to find out what 2.37 x 4.19 is equal to (and how could you use this to work out 237 000 x 4 190 000  or  0.00237 x 0.000 0419 ?).

c) Repeat this process three times, each time using numbers in the above table, and then adding the powers and finding what number this corresponds to. Check your answers using your calculator.

Note: If you're interested in understanding how these log tables were calculated in an era before calculators, see the "History of Logs" section below.

### With Calculators, why do we still use logs today?

Plot the data below in any spreadsheet graphing software of your choice: Geogebra, Autograph, TiNspire or Casio, Desmos etc. You can copy and paste it directly from the below spreadsheet:

What problem do you notice?

Your divisibility tests may help. Log tables use powers of 10, but you could equally use powers of 2 or 5 or 9 . . . or any number you prefer.

#### You understand what a Log is? Put your knowledge to the test:

Test 1: Log(100) is asking a question - if you can translate the mathematical notation and ask yourself the question, the answer becomes obvious. Translate the mathematics!

Log(10 000)=

104 = 10 000

Log(0.001)=

10-3 = 0.001

Log(100)=

102 = 100     Any number to the power of zero = 1 Negative powers are the "reciprocal" of the positive power For all questions from here onwards, tick ALL the answers that are correct (there may be more than one).

Convert both 100 and into powers of ten. Then when you multiply powers you know to add the powers.  Laws of powers/indices: multiply powers when raising one power to the power of another e.g. (10a)b = 10ab

Log[10m]=

The power of 10 is equal to 'm' in this question Negative powers are the "reciprocal" of the positive power This is the same question as Log(0.001): 0.001 = 10-3 When you divide powers you subtract them: OR multiplying powers you add them:  Negative powers are the "reciprocal" of the positive power. When dividing powers, you subtract the powers. Convert both into a power of 10 and then add the powers: Total Score:

#### Test 2: Logs NOT to the base 10 There could be more than one answer in the questions that follow.

24=16 9²=81 33=27   Ln means log to the base 'e' where For the following questions you need to find the value of the letter 34 = 81 23 = 8 5-3 =  5-2 =   Total Score:

#### Gapminder (UN data)

Many of the graphs concerning human well-being, from the Gapminder website, use a logarithmic scale. Can you explain why the x-axes below is a logarithmic scale but not the y-axes? #### The Richter Scale

You can find a range of "Earthquake" apps on android and itunes that will offer you alerts on earthquake activity around the world, as well as some offering you data on earth tremor activity detected by your phone. Here's a couple to get you started:

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For an explanation of how the Richter scale is calculated, watch the first three minutes of this video from the Khan Academy

#### The Decibel scale

There's a range of decibel meters available both on android and itunes, that provide great opportunities for further investigation and exploration of logarithmic scales and cross-curricula (DT, Science & Geography) projects:

Click on the images below to get a feel for the noise level to expect for a given number of decibels:

#### Modelling very varied Data

In the Influenza pandemic graph below, the y-axis is a logarithmic scalewhat base is it using?

What does each unit on the y-axis represent? http://en.wikipedia.org/wiki/File:Influenza-2009-cases-logarithmic.png

#### History of Logs

Two fantastic articles from the "Math Forum": How were the first log tables calculated (given there were no calculators)? History of Logarithmic tables

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