SOLUTION: y=1000(2raised to the -.1x) if x=10, what is y=? then, if y=250, what is x=?

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Question 367073: y=1000(2raised to the -.1x)
if x=10, what is y=?
then, if y=250, what is x=?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
y=1000%282%5E%28-0.1x%29%29
It help to rewrite this with a positive exponent. In general, a%5E%28-p%29+=+1%2Fa%5Ep. so your expression can be rewritten as:
y=1000%281%2F2%5E%280.1x%29%29

if x = 10...
y=1000%281%2F2%5E%280.1%2810%29%29%29
which simplifies as follows:
y=1000%281%2F2%5E1%29
y=1000%281%2F2%29
y+=+500

if y = 250...
250=1000%281%2F2%5E%280.1x%29%29
Divide both sides by 1000:
250%2F1000=%281000%281%2F2%5E%280.1x%29%29%29%2F1000
which simplifies to:
1%2F4=1%2F2%5E%280.1x%29
This is a proportion so we can cross multiply:
1%2A2%5E%280.1x%29+=+4%2A1
or
2%5E%280.1x%29+=+4
The shortest way to solving for x here is to recognize that both sides of the equation are powers of two:
2%5E%280.1x%29+=+2%5E2
(Note: If both sides had not both been powers of the same number, then we would have to use logarithms to solve for x.)
The only way for powers of 2 to be equal is if their exponents are equal. So:
0.1x+=+2
And finally we can multiply both sides by 10:
10%280.1x%29+=+10%282%29
which simplifies to:
x+=+20