SOLUTION: Given: {{{ log (a,x) = 5 }}} {{{ y = log (a,y) }}} Find the value of x.

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Given: {{{ log (a,x) = 5 }}} {{{ y = log (a,y) }}} Find the value of x.      Log On


   

Question 1194342: Given:
+log+%28a%2Cx%29+=+5+
+y+=+log+%28a%2Cy%29+
Find the value of x.
Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!


There are infinitely many solutions, here's one, approximately:



a=0.5, x=0.03125, y=0.642286



log%280.5%2C0.03125%29+=+ln%280.03125%29%2Fln%280.5%29+=+5



0.641186+=+log%280.5%2C0.641186%29+=+ln%280.641186%29%2Fln%280.5%29+=+0.6411851696+, approximately



----------------------------



here's another, approximately:



a=1.2, x=2.48832, y=1.25773



log%281.2%2C2.48832%29+=+ln%282.48832%29%2Fln%281.2%29+=+5



1.25773+=+log%281.2%2C1.25773%29+=+ln%281.25773%29%2Fln%281.2%29+=+1.257714737+, approximately.



Edwin

Answer by ikleyn(52790) About Me  (Show Source):
You can put this solution on YOUR website!
.

You are given two equations (system of two equations)


    log%28a%2C%28x%29%29 = 5        (1)

    log%28a%2C%28y%29%29 = y        (2)



Divide equation (1) by equation (2).  You will get


    log%28a%2C%28x%29%29%2Flog%28a%2C%28y%29%29 = 5%2Fy.      (3)


Apply the change-of-base formula for logarithms to the left side of equation (3).  Instead of equation (3), you will get then


    log%28y%2C%28x%29%29 = 5%2Fy.      (4)



It means that


    x = y^(5/y).        (5)


It is the sough expression of x via y,  with parameter  "a"  highlight%28excluded%29.


Expression (5) is your answer.   It allows calculate the value of x for any given positive value of y.

Solved and thoroughly explained.


///////////////


My comment after solving the problem: 

    (1)  The problem is highlight%28posed%29 highlight%28incorrectly%29.


         It asks  " Find the value of x. "

         But in this problem, there is no a unique single specific "value of x"
            and,  THEREFORE,  there  is  NOTHING  to  FIND  in this sense.

         There are infinitely many values x and y satisfying given equations.


    (2)  The correct formulation of the problem should be


            +--------------------------------------------------+
            |    Exclude parameter "a" from given equations    |
            |         and express x as a function of y.        |
            +--------------------------------------------------+


         It was EXACTLY what I did in my solution.