Logarithms!

1. Logarithms! Objective: To understand the history of logarithms and how to use them.

2. Warm up 1.) a. 4-4 b. (x3) 3 c. (-x)9 2.) Graph y=4x from (-3, 3)

3. History of logs • In 1614, John Napier discovered the logarithm which made it possible to perform multiplications and divisions by addition and subtraction. (ie: a*b = 10^(log(a)+log(b)) and a/b = 10^(log(a)-log(b)).)                                     Slide Rule • The scale started at one because the log of one is zero. Two numbers could be multiplied by measuring the distance from the beginning of the scale to one factor with a pair of dividers, then moving them to start at the other factor and reading the number at the combined distance.

4. Notes • Finding the index of a logarithmThis is easy if you follow these rules:For numbers with more than 1 digit before the decimal point (10 and above), the index will always be one less than the number of digits before the decimal point. For example, the index of 245.211 is 2, the index of 24521.1 is 4 and the index of 24.5211 is 1.For numbers with just 1 digit before the decimal point (1 to 9), the index will be 0. For numbers less than 1, then index will be one more than the number of zeros between the decimal point and the first significant figure. For example, the index of 0.245211 is (bar) 1, the index of 0.00245211 is (bar) 3 and the index of 0.000245211 is (bar) 4.

5. Notes • For numbers with just 1 digit before the decimal point (1 to 9), the index will be 0. For numbers less than 1, then index will be one more than the number of zeros between the decimal point and the first significant figure. For example, the index of 0.245211 is (bar) 1, the index of 0.00245211 is (bar) 3 and the index of 0.000245211 is (bar) 4.

6. Notes • Finding the mantissa of a logarithmUse the numbers at the far left of the table to give the first two significant figures of the number. If there are more than two digits in the number, follow across the table - the column headings give the third digit of the number. The numbers given in the log table represent the mantissa part of the answer.

7. Notes • Example: To find log 45, follow down the numbers on the far left of the table until you reach 45. The number in the column headed 0 will be the mantissa - the index is 1 (see above) - so log 45 = 1.6532.To find log 456, follow down the numbers on the far left of the table until you reach 45 and take the number in column 6. The index is 2, so log 456 = 2.6590.

8. Vocabulary • Logarithm- the exponent of the power to which a base number must be raised to equal a given number; log: 2 is the logarithm of 100 to the base 10 (2 = log10 100). • Exponent- a symbol or number placed above and after another symbol or number to denote the power to which the latter is to be raised: The exponents of the quantities x n , 2 m , y 4 , and 3 5  are, respectively, n, m, 4, and 5.

9. Log Table • http://www.ask.com/bar?q=log+table&page=1&qsrc=2891&dm=all&ab=0&u=http%3A%2F%2Fwww.sosmath.com%2Ftables%2Flogtable%2Flogtable.html&sg=58614rjqbm7dZ5Yovid5ulfI4IfgWdEaKKlL7pyslI8%3D&tsp=1271693663376

10. Video! • http://www.youtube.com/watch?v=NB_lHeBY26Y

11. Citations • Slide Rules". The Museum of HP Calculators. 4/20/10 <http://www.hpmuseum.org/sliderul.htm>. • Unknown, . "Logarithms." edhelper. Unknown, 2010. Web. 20 Apr 2010. <http://edhelper.com/Logarithms20.htm>. • "Log Tables." Math. DOMEX E-DATA PVT. LTD, 2009. Web. 20 Apr 2010. <http://www.scenta.co.uk/tcaep/maths/number/logtab/index.htm>.