LOGARITHMIC FUNCTIONS

LOGARITHMIC FUNCTIONS

*Steps how to convert EXPONENTIAL FORM into LOGARITHMIC FORM.

1 - Replace f(x) with y.

2 - Reverse the rules of x and y.

3 - Solve for y in terms of x.

4 - Replace y with f ^-1(x).

Example:
Find the inverse of  f(x)=a^x using the four step method.

Exponential form:                Logarithmic form:      

 f(x)=a^x                                         y=loga^x                     

y=a^x                                               y=exponent

x=a^y                                              a= base

                                               x=answer  

 " Rules of logarithmic "        "Remember"

               x>0                                   y=exponent

               a>0                                   a= base

                a1                                  x=answer 

*Examples for solving Logarithmic form:

Note: Some of the calculators are based only on log₁₀

.
 (x>0) - rules of logarithmic                                            (x>0)

log₁₀(-5)=Error                                log₁₀0=Error


     (x<0)                                                                        (x>0)

log_101/2= 0.301029995                  log<sub>101=0


*The next question that i will show you is different base.</sub>

 log<sub>232="5" 


*The base that i used is 2, So how will I know the answer?</sub>

 _{We just need to find, what is visible number for 32 so what i did is "} 2⁵_"   because it is equal to "32"

 More examples: 

   

log₃₂₇₌₃                log₄₁₆₌₂

_{*Example for converting Logarithmic form into exponential form:}

  _{note: Multiply the base to the exponent and write down the answer.}

 _{Question                                      Answer}

_a.)log_{10100 =2}                     10²=100

   Base = 10

    Exponent = 2

    Answer = 100

b.)4 = log₃₈₁                       3⁴=81  

   Base = 3

    Exponent = 4

    Answer =81

c.) log_{39 =2}                             3²=9        

   Base = 3

    Exponent = 2

    Answer = 9

*Example for converting exponential form into Logarithmic form:

 Note: easier to convert is "LOG a^y=x"

  Questions                                                  Answer

a.)7^u=v                                  log_{7v =u} 

     a = 7

     y = v

     x = u

 b.) 5²=25                                    log_{525 =2}  

     a = 5

     y=25

     x=2

  
 
c.)27^2/3=9                             log_{279 =2/3
   a = 27
     y = 9
    x = 2/3}



*Example for evaluating Logarithmic form
 Note: First Convert it into exponential form then find a number that visible by the answer so the base will be the same then cancel both base and you'll find the answer.

Question                                                           Answer
a.) log_{5125 =x}                             5^x = 125
                                                5^x= 5³
                                                          x=3


b.) log31/9 =x                               3^x = 1/9        
                                                  3^x = 9^-1    
                                                  3^x = 3^-2   
                                                    x=-2




c.)log6216                                   6 = 216  
                                                    6 = 6³
                                                        =3





That's all folks enjoy!!!..