ajaz_eco204_2009_chapter_0

ajaz_eco204_2009_chapter_0 - University of Toronto,...

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University of Toronto, Department of Economics, ECO 204 2009 2010 S. Ajaz Hussain ECO 204 2009 2010 S. Ajaz Hussain (Draft) Chapter 0: Math Preliminaries Please help improve the course by sending me an e mail about typos or suggestions for improvements In this “chapter” you will review logs and derivatives. You may also want to review these out of your Math 133 textbook and notes (assuming you haven’t sold, burnt or shredded these!). Logarithms (Logs) Almost everyone has bad memories of logs. In my opinion this is because logs are forced onto hapless students without showing them where logs come from or why they’re so useful. So let’s start by seeing where logs come from. Look at the following numbers: 3 ൌ9 2 ൌ8 5 ൌ 125 10 ൌ 100 3 ry log e base rais So, each of the statements above can be re expressed as: 2 ൌ8֞ log 8ൌ3 5 ൌ 125 ֞ log 125 ൌ 3 10 ൌ 100 ֞ log ଵ଴ 100 ൌ 2 See how logs work? When you We can re express of these numbers through logs. For example, the statement can be re expressed as: log 9ൌ2 . Note how this log has the “base” 3 (and indeed eve has a “base”) where th ed to the power 2 gives you 9. Put simply, saying log 9ൌ2 is to say that 3 and vice versa. 3 ֞ log 9ൌ2 say: 1 ECO 204 (Draft) Chapter 0
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University of Toronto, Department of Economics, ECO 204 2009 2010 S. Ajaz Hussain log ௕௔௦௘ ܽൌܿ You really mean: ܤܽݏ݁ ൌܽ and vice versa. useful properties. Some of these are for constants ܽ and ܾ and any base: log ቀ ܽ ܾ logሺܾܽሻൌlogܽ൅logܾ Logs have some ቁൌlogܽെlogܾ log ܽ ൌܾlogܽ logሺܽേܾሻ്logܽേlogܾ In the old days, it was faster and cheaper for a computer to add than to multiply (divide) two numbers. Hence, if a programmer had to multiply (divide) two very large numbers ܽ,ܾ she ould program the computer to first add (subtract) the log of these two numbers and then F no tion we’ll see again and again. You may remember from ECO 100 that a demand curve can be linear or non linear. For example: In ECO 204 and ECO 220 it is often use ke a non linear equation and make it “linear”. Here is how: ܳൌ5ܲ ିଷ ௬௜௘௟ௗ௦ w recover ܾܽ ( ܽ/ܾ ) by taking the anti log. When we do derivatives of logs below you will see why logs are also useful for calculating changes. or w, let’s look at an applica Linear: ܳൌ3െ2ܲ Non linear: ିଷ ful to ta ۛۛۛሮ log ܳ ൌ logሺ5ܲ ିଷ ௬௜௘௟ௗ௦ ሱۛۛۛሮ log ܳ ൌ log 5 ൅ log ܲ ିଷ ௬௜௘௟ௗ௦ ሱۛۛۛሮ log ܳ ൌ log 5 ൅ ሺെ3ሻlog ܲ ௬௜௘௟ௗ௦ ሱۛۛۛሮ log ܳ ൌ log 5 െ 3 log ܲ Look closely: this is the equation a straight line where the ݕ variable is log ܳ and the ݔ variable is log ܲ . Observe that the graph of: of 2 ECO 204 (Draft) Chapter 0
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University of Toronto, Department of Economics, ECO 204 2009 2010 S. Ajaz Hussain ܳൌ5ܲ ିଷ of: log 5 െ 3 log ܲ with log ܳ on the ݕ axis and log ܲ on the ݔ axis is linear. As such, it is common to call such
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This note was uploaded on 05/02/2011 for the course ECO 204 taught by Professor Hussein during the Fall '08 term at University of Toronto.

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ajaz_eco204_2009_chapter_0 - University of Toronto,...

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