SOLUTION: if (125/27)^x = the cubic root of 0.6 then the value of x is _______

Algebra ->  Square-cubic-other-roots -> SOLUTION: if (125/27)^x = the cubic root of 0.6 then the value of x is _______       Log On


   

Question 202363: if (125/27)^x = the cubic root of 0.6 then the value of x is _______


Found 3 solutions by Theo, ikleyn, greenestamps:
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
your original equation is:
%28125%2F27%29%5Ex+=+%28.6%29%5E%281%2F3%29
by the laws of logarithms, this can only be true if and only if
log%28%28125%2F27%29%2C%28%28.6%29%5E%281%2F3%29%29%29+=+x+
we can convert this log with a base of (125/27) to a log with a base of 10 so that we can solve it using the calculator.
the logarithm base conversion formula is:

our formula now becomes:
+log%2810%2C%28%28.6%29%5E%281%2F3%29%29%29+%2F+log%2810%2C%28125%2F27%29%29+ = x.
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now:
log%2810%2C%28%28.6%29%5E%281%2F3%29%29%29 is the same as %281%2F3%29%2A%28log%2810%2C%28.6%29%29%29.
also:
+log%2810%2C%28125%2F27%29%29+ is the same as +log%2810%2C125%29+-+log%2810%2C27%29+
so the equation becomes:
x = %28%281%2F3%29%2A%28log%2810%2C%28.6%29%29%29%29 / +%28log%2810%2C125%29+-+log%2810%2C27%29%29+
you can now use your calculator to solve this.
the equation becomes:
x = %28%281%2F3%29%2A%28-.22184875%29%29%2F%28%282.096910013%29-%281.431363764%29%29
which becomes:
x = %28-0.073949583%29%2F%28.665546249%29
which becomes:
x = -0.1111111111111
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to prove your answer is correct, substitute it in the original equation to get:
%28125%2F27%29%5E%28-0.111111111111%29+=+%28.6%29%5E%281%2F3%29
this becomes:
0.843432665+=+.843432665
which is true.
note that .843432665 = (.6)^(1/3)
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Answer by ikleyn(53250) About Me  (Show Source):
You can put this solution on YOUR website!
.
if (125/27)^x = the cubic root of 0.6 then the value of x is . . .
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


        This problem can be solved by applying brute force,  as tutor  @Theo did it.

        But it can be solved elegantly,  if to look at the numbers and to think a bit. 


Notice that the ratio  125%2F27  in the left side is nothing else as  %285%2F3%29%5E3.

In the right side,  0.6 = 6%2F10 = 3%2F5.


This suggests that the left side is  %285%2F3%29%5E%283x%29,  while the right side is  %285%2F3%29%5E%28-1%2F3%29.


So, our equation is

    %285%2F3%29%5E%283x%29 = %285%2F3%29%5E%28-1%2F3%29.


This implies

    3x = -1%2F3,  which gives  x = -1%2F9.


ANSWER.  The solution to this equation is  x = -1%2F9.

Solved, and the exact solution is found.

Notice that -1%2F9 = -0.111111111111111...


Happy learning  ( ! )

This problem is nice  !     It is  highlight%28highlight%28specially%29%29  designed so that you would get the solution in  highlight%28highlight%28PRECISELY%29%29  highlight%28highlight%28this%29%29  form
and would get the taste of mathematical discovery.



Answer by greenestamps(13240) About Me  (Show Source):