logarithm cheat sheet, Cheat Sheet of Mathematics

Download logarithm cheat sheet and more Cheat Sheet Mathematics in PDF only on Docsity! Properties of Exponents and Logarithms Exponents Let a and b be real numbers and m and n be integers. Then the following properties ofexponents hold, provided that all of the expressions appearing in a particular equation arede ned. 1. aman = am+n 2. (am)n = amn 3. (ab)m = ambm 4. aman = amn, a 6= 0 5. a b m = ambm , b 6= 0 6. am = 1 am , a 6= 0 7. a 1n = npa 8. a0 = 1, a 6= 0 9. amn = npam = npa
m where m and n are integers in properties 7 and 9. Logarithms De nition: y = loga x if and only if x = ay, where a > 0.In other words, logarithms are exponents. Remarks:  log x always refers to log base 10, i.e., log x = log10 x.  lnx is called the natural logarithm and is used to represent loge x, where the irrationalnumber e  2:71828. Therefore, lnx = y if and only if ey = x.  Most calculators can directly compute logs base 10 and the natural log. For any other base it is necessary to use the change of base formula: logb a = ln aln b or log10 alog10 b . Properties of Logarithms (Recall that logs are only de ned for positive values of x.) For the natural logarithm For logarithms base a1. ln xy = lnx+ ln y 1. loga xy = loga x+ loga y2. ln xy = lnx ln y 2. loga x y = loga x loga y3. ln xy = y
lnx 3. loga xy = y
loga x4. ln ex = x 4. loga ax = x5. elnx = x 5. aloga x = x Useful Identities for Logarithms For the natural logarithm For logarithms base a1. ln e = 1 1. loga a = 1, for all a > 02. ln 1 = 0 2. loga 1 = 0, for all a > 0 1