EXPONENTS AND LOGARITHMS

e e is a mathematical constant ≈ 2.71828… Commonly used as a base in exponential and logarithmic functions: exponential function – ex natural logrithm – logex or lnx follows all the rules for exponents and logs E is also limit of (1+1/n)n as n becomes large, is the sum of 1/i for i=0 to infinity

EXPONENTS an where a is the base and x is the exponent an = a · a · a · … · a e3 = e * e * e a1 = a e1 = e a0 = 1 e0 = 1 a-n = 1 a n e-2 = 1 e 2

EXPONENTS Using your calculator: 10x: base 10 ex: base e yx: base y Try: e2 = e1.5 = FOR THOSE CALCULATORS that have y to the x

LAWS OF EXPONENTS The following laws of exponents work for ANY exponential function with the same base

LAWS OF EXPONENTS aman = am+n e3e4 = e3+4 = e7 exe4 = ex+4 a2e4 = a2e4 Try: e7e11 eyex

LAWS OF EXPONENTS (am)n = amn (e4)2 = e4*2 = e8 (e3)3 = e3*3 = e9 (108)5 = 108*5 = 1040 Try: (e2)2 (104)2

LAWS OF EXPONENTS (ab)n = anbn (2e)3 = 23e3 = 8e3 (ae)2 = a2e2 Try: (ex)5 2(3e)3 (7a)2

LAWS OF EXPONENTS a b n = a n b n e 3 2 = e 2 3 2 = e 2 9 Try: e 2 3 2a e 3

LAWS OF EXPONENTS a m a n = a m−n ≡ 1 a n−m e 7 e 8 = e 7−8 = e −1 = 1 e Try: e 4 e 2 3e 2 e 2

LAWS OF EXPONENTS Try these: x −3 y 4 a 3 b −2 a −5 b 7 5 a 3 b 3a b 2 3

LOGARITHMS The logarithm function is the inverse of the exponential function. Or, to say it differently, the logarithm is another way to write an exponent. Y = logbx if and only if by = x So, the logarithm of a given number (x) is the number (y) the base (b) must be raised by to produce that given number (x)

LOGARITHMS Logarithms are undefined for negative numbers Recall, y = logbx if and only if by = x blogbx = x eloge2 = eln2 = 2 (definition ) logaa = 1 logee = lne = 1 (lne = 1 iff e1 = e) loga1 = 0 loge1 = ln1 = 0 (ln1 = 0 iff e0 = 1) Substitute v and u in above definitions

LOGARITHM Using your calculator: LOG: this is log10 aka the common log LN: this is loge aka the natural log x < 1, lnx < 0; x > 1, lnx > 0 Try: ln 0 = ln 0.000001 = ln 1 = ln 10 =

LAWS OF LOGARITHMS logb(xz) = logbx + logbz ln(1*2) = ln1 + ln2 = 0 + ln2 = ln2 ln(3*2) = ln3 + ln2 ln(3*3) = ln3 + ln3 = 2(ln3) Try: ln(3*5) = ln(2x) = Try both ways: ln15 = 2.7080502 and ln3 + ln5 = 1.0986123 + 1.6094379 = 2.708…

LAWS OF LOGARITHM logb x z = logbx – logbz loge 2 3 = ln2 – ln3 Try: ln 2 7 = ln x 7 = Try both ways 2/7 = 0.2857143 ln0.28… = -1.2527630 and ln2 – ln7 = 0.6931472 – 1.9459101 = -1.25…

LAWS OF LOGARITHMS logb(xr) = rlogbx for every real number r loge(23) = 3ln2 loge(32) = 2ln3 Try: loge42 = logex3 = ln3x =

ln and e Recall, ln is the inverse of e Try: x = 2 x = 0.009 x lnx elnx 1 1.5 0.40546 e0.40546 = 1.5 3 1.09860 e1.09860 = 3

EXAMPLES OF LOGARITHMS Try: w/o calculator lne5 rewrite in condensed form: 2lnx + lny +ln8 3ln5 – ln19 expand: ln10x3

RADICALS a is the radicand n is the index of the radical is called a radical a is the radicand n is the index of the radical is the radical sign by convention and is called square root

LAWS OF RADICALS Laws of radicals follow the laws of exponents: Try:

SCIENTIFIC NOTATION Numbers written in the form a x 10b when b is positive – move decimal point b places for the right when b is negative – move decimal point b places to the left Reverse the procedure for number written in decimal form Follows the laws of exponents

EXAMPLES OF SCIENTIFIC NOTATION 1,003,953.79 1.00395379 x 106 -29,000.00 -2.9 x 104 0.0000897 8.97 x 10-5