SOLUTION: find the value of a if 7^a - 7^a-5 = 117642 sqrt 7

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Question 1186981: find the value of a if 7^a - 7^a-5 = 117642 sqrt 7
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
.
find the value of a if 7^a - 7^(a-5) = 117642 sqrt 7
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The starting equation

    7%5Ea+-+7%5E%28a-5%29 = 117642%2Asqrt%287%29


is EQUIVALENT to

    7%5E%28a-5%29%2A%287%5E5-1%29 = 117642%2Asqrt%287%29,

    7%5E%28a-5%29%2A16806   = 117642%2Asqrt%287%29,


and then, canceling the common factor  16806 in both sides

    7%5E%28a-5%29 = 7%2Asqrt%287%29.


It implies  

    a - 5 = 3%2F2,


hence

    a = 5 + 3%2F2 = 6 1%2F2 = 6.5.    ANSWER

Solved.



Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
7%5Ea+-+7%5E%28a-5%29+=+117642sqrt%287%29

7%5Ea+-+7%5Ea%2A7%5E%28-5%29+=+117642%287%5E%28%221%2F2%22%29%29

Factor out 7a

7%5Ea%2A%281+-+7%5E%28-5%29%29+=+117642%287%5E%28%221%2F2%22%29%29

Divide both sides by 71/2

%287%5Ea%2A%281+-+7%5E%28-5%29%29%29%2F7%5E%28%221%2F2%22%29+=+117642

%287%5Ea%2F7%5E%28%221%2F2%22%29%29%281+-+7%5E%28-5%29%29+=+117642

7%5E%28a-%221%2F2%22%29%281+-+7%5E%28-5%29%29+=+117642

7%5E%28a-%221%2F2%22%29%281+-+1%5E%22%22%2F7%5E5%29+=+117642

7%5E%28a-%221%2F2%22%29%281+-+1%5E%22%22%2F117642%29+=+117642

7%5E%28a-%221%2F2%22%29%281+-+1%2F16807%29+=+117642

7%5E%28a-%221%2F2%22%29a%2816807%2F16807+-+1%2F16807%29+=+117642

7%5E%28a-%221%2F2%22%29%2816806%2F16807%29+=+117642

Multiply through by 16807

7%5E%28a-%221%2F2%22%29%2816806%29+=+1977209094

Divide both sides by 16806

7%5E%28a-%221%2F2%22%29=117649

Take logs (base 10) of both sides

log%287%5E%28a-%221%2F2%22%29%29=log%28117649%29

%28a-1%2F2%29%2Alog%287%29=5.07058824

a-1%2F2=5.07058824%2Flog%287%29

a-1%2F2=6

a=6%2B1%2F2

a=13%2F2

Edwin