In this investigation I will be examining logarithms and their bases. The purpose of examining logarithmic bases is to find patterns between the bases, its exponents and the product.

Figure 1.1 shows 4 graphs and their points of intersection.

Y= s(x), where s(x) =

Y= t(x), where t(x) = log28

Y = u(x), where u(x) = log168

Y = v(x), where v(x) = log88

These graphs support the expression  and their points of intersection also fit the theory mentioned above.

Figure 1.2 shows 4 graphs and their points of intersection

Y = f(x), where f(x) =

Y = h(x), where h(x) = log381

Y = q(x), where q(x) = log981

Y = r(x), where r(x) = log2781

 These graphs also support the expression  and the points of intersection also fit the theory mentioned above.

Figure 1.3 shows 4 graphs and its points of intersection

Y = f(x), where f(x) = log525

Y = g(x), where g(x) =

Y = h(x), where h(x) = log12525

Y= s(x), where s(x) = log 2525

These graphs support the expression  and its point of intersection also fit the theory mentioned above.

I will now calculate the values for other sequences and try to find a pattern on how to obtain the answers. All answers are written in the form , where p, q  Z.

1. log464                  log864                    log3264

   = 3                        = 2                          =

 

2. log749                  log4949                  log34349

    = 2                       = 1                         =

3. log125               log125             log125

    = - 3                     = -1                        = -

4. log8512                log2512                log16512

    = 3                       = 9                        =

 

After analyzing the above sequences, I noticed a pattern between the first two answers and the third answer. To find the third answer it is necessary to multiply the first two answers. The value after multiplying the first two answers produces the numerator for the third answer. To find the denominator for the third answer it is necessary to add up the first two answers. Thus, the third answer is usually written in the form  which in some cases can be simplified further. To prove this theory I will create two more sequences which follow the same pattern as the sequences above. Then I will evaluate the terms to prove my theory.

1.  log6216              log36216                log216216

     = 3                      = 1.5 or             = 1

                       

2.  log9729              log3729                 log27729

     = 3                     = 6                         = 2

                       

The pattern shown above can be generalized for most logarithms. So, I will create a general statement for logarithms following this pattern.

Let logax = c and logbx = d

So, logabx =

Basic exponent and logarithm properties can be used to derive the general statement.

   

          Let logax = c                      Let logbx = d

By using a logarithm property we can convert these logs into exponents:

                 ac = x                         bd = x

               

                 (ac)d = xd                    (bd)c = xc

Then we can multiply both equations:

               acd = xd                  bdc = xc

           

                (ab)cd = xc+d

Now we can use log properties to simplify the equation:

                 cd log(ab) = c+d log(x)

                 

                 

                    = logabx

So, the general statement is found  = logabx  

To test the validity of my general statement I will use other values of a, b and x to see if my results match the general statement.

1. log164096               log44096                 log644096

   = 3                           = 6                          

2. log121728              log61728                 log721728

    = 3                         = 4.16                    

3. log27729               log3729                    log81729

    =2                         = 6                          

4. log2401               log2401               log2401

    = -2                           = -4                        

5. log20736               log20736                log20736

    = -2                             = -4                            

After testing the validity of the general statement, I realized that the statement does not work for all numbers. There are some limitations for the values of a, b and x. the first limitation is that all values of a, b and x must be greater than 0. The values must be greater than 0 because it is not possible to calculate the log of a negative number or 0. In other words the values of a, b and x must be all real numbers. Another limitation to this statement is that the sum of a+b must be greater than 0. This restriction applies because if the values of a+b are less than 0 then you have an error in the domain. The final restriction is that the value of a can not be. This applies to the general statement because it is impossible to divide a number by 0. Thus, these are all the limitations on the general statement.

Throughout this portfolio I have investigated the patterns between logarithms and their bases. To find these patterns I looked at different sequences of logarithms. Then I found relationships between the values of the first two terms and how that value relates to the third term. By following this process I was able to find a general statement and test its validity. Then I described the limitations as well as the scope of the general statement. So, by completing this portfolio I was able to understand a lot about logarithms, their bases and the relationships between them.

Fig. 1.1

Fig. 1.2

Fig. 1.3