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à DEFINITION: For x>0 and 0< a≠ 1,    y=logax, if and only if x=ay

Every function of the form f(x)=ax,         a>0, a≠1    passes the Horizontal Line Test and therefore must have an inverse.—This inverse function is called the logarithmic function with base a.

 

NOTE: WHEN READING y=logax, READ LOG OF X TO THE BASE A.

              A LOGARITHM IS AN EXPONENT

 

à The function f(x)= logax is called the logarithmic function with base a.

            LOGARITHMIC FORM: y=logax

            COMMON LOG: base 10

            NATURAL LOG: base e

                                                ↓

The function defined by f(x)=logex = lnx, x>0 is the natural logarithmic function.

à  EVALUATING LOGARTITHMS:

 

            1. log232          2. log327          3. log31            4. log22

 

à  SOLUTIONS:

 

1.      log232=5   because 25= 32.

2.      log327=3   because 33= 27.

3.      log31=0     because 30= 1.

4.      log22=1     because 21=2.

 

à REWRITING LOGARITHMIC FUNCTIONS WITH DIFFERENT BASES

           

            BASE B                       BASE 10                     BASE e

 

            logax = logbx                 logax = log10x    logax = lnx

                        logba                              log10a                         lna

 

 

 

à  CHANGING BASES USING COMMON LOGARITHMS

 

log430 = log1030                       1.47712                       2.4534

               log104           à            0.60206      à

 

 

à  CHANGING BASES USING NATURAL LOGARITHMS

 

            log430 = ln30            à        3.40120        à              2.4535

                           ln4                           1.38629

                                                           

 

 

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