Properties of Logarithms Section 9.0B   Find x for                 log332 = x        rewrite                     3x = 32   therefore          x = 2         Log525 = x                          5x =25                      x = 2         Ln e3 = x                  ex = e3                      x = 3         Log 10 = x               10x = 10                   x = 1   Inverse properties:         log10x = x         ln ex = x   Do not confuse log with ln.           log ex x  and          ln 10x x ,   More inverses (the functions undo each other).    For example          ( )2 = x           elnx = x           10logx = x   Example 1:         Solving exponential equations.   a)         10x = 3                                                Take the common log of both sides Log10x = log 3                                   Inverse property X = log 3 = 0.47712155…               Use calculator   b)         2e3x = 5          the base is e, but before taking ln of both sides, isolate e3x             e3x  = 5/2                    take ln of both sides             ln e3x = ln (5/2)           Inverse property             3x = ln (5/2)                Solve for x             x = ln(5/2) = 0.305430244                      3   Property:  Exponent becomes multiplier         log Ar = r log A                                ln Ar = r ln A   Proof:   Let                                        log A = y rewrite                                 10y      = A raise both sides to n                      (10y)n = An take log of both sides                    log 10ny = log An Inverse Property                            ny = log An                   but          y = log A               (given) Therefore                                        n(y) = n(log A) Example 3:               Solve           32x = 5                          The base is neither 10 nor e, but we can still take common log of both sides and use multiplier rule.             Log 32x = log 5             2xlog 3 = log 5              x  = log5/2log3 = 0.73248676   Division becomes subtraction property.   log A/B = log A – log B   ln A/B = ln A – ln B   Note:     log A/log B log A – log B   Proof:    Let                                        a = log A         b = log B Rewrite                                  10a = A           10b = B Divide                                     A/B = 10a/ 10b Property of exponents             A/B = 10a – b Take log of both                      log A/B = log 10a – b Inverse Property                      log A/B = a – b Given                                       log A/B = log A – log B   EX 5:   Rewrite            a)          log 5x/3                  without fraction           log 5x – log 3   b)        log 3 – log 2x         as a single log          log 3/2x   Multiplication becomes addition property.           log (AB) = log A + log B                     ln (AB) = ln A + ln B   Extra Credit: Prove this property.   Exercise 6:   Rewrite            a)  log 4x                     as separate logs         Log 4 + log x   b)  log 5 + log 3x         as a single log         log 15x   Exercise 7:  Using all three properties (multiplier, division, addition) solve the following.   Log x + log 5 = 3                               Addition property Log 5x = 3                                          Rewrite into exponential form 103 = 5x                                              Simplify 1000 = 5x                                           Solve for x 1000/5 = x = 200 Back to Exponenials and Logarithms Main Page Back to the Survey of Math Ideas Home Page e-mail Questions and Suggestions