Expert: Steve Holleran - 8/28/2008

Question
QUESTION: Hello, we are currently changing our data series of rainfall data to incorporate climate change.  The equation for the 'new' rainfall is:
y = ax^b where
y = new rainfall
a = unknown constant
x = old rainfall - 0.5
b= unknown constant

Is there anyway to solve this for y without knowing the values of a and b?  I tried to use the log rules but ended up in circles with that.

Thanks

ANSWER: Hi Joanne,

You could do this using logs, but you could also use roots:

         y = a * x^b  -->  (y/a) = x^b

now take the "b th" root of each side:

         x = b th rt(y/a)

For example, if it were y = 3 * x^6

then   x = 6 th rt(y/3).

Steve

---------- FOLLOW-UP ----------

QUESTION: Thanks for that.
I understand how you have used the roots, but is there anyway I can solve (perhaps using a numerical method) for y when a and b are unknown?

ANSWER: Hi Joanne,

I am not aware of a numerical method to do this, and I don't see how you can get a value for y if a and b are unknown.

Steve

---------- FOLLOW-UP ----------

QUESTION: Hi Steve, I was beginning to suspect the same myself so it was great to have you confirm that and save me hours of pondering and getting no where.  
I now have a variation of the equation to solve.
Assuming I know all variables (apart from x), how do you solve this equation:

y = (a*(b-c)^x)+0.5

I got to:
log y = log ((a*(b-c)^x)+0.5)
but then I got stuck with how to deal with the +0.5.

Can you help?
Cheers,
Joanne

Answer
Hi Joanne,

Okay, if you're trying to solve for x, then I would move the 0.5 over at the very first step:

         y = a * (b-c)^x + 0.5

         y-0.5 = a * (b-c)^x

         (y-0.5)/a = (b-c)^x

Now take logs:    log[(y-0.5)/a] = log(b-c)^x = x * log(b-c)

so          x = log[(y-0.5)/a] / log[b-c]

I hope this is what you wanted.
Steve