anonymous
  • anonymous
How to find the variables a and b, in the function y = ax^b, using algebraic methods?
Mathematics
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anonymous
  • anonymous
How to find the variables a and b, in the function y = ax^b, using algebraic methods?
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
I already have a table of values for x and y. I need to use an algebraic method other than linear regression to model the function.
myininaya
  • myininaya
use the points to find a and b for example say you have (1,4) and (2,16) on your graph then we have \[4=a(1)^b \text{ and } 16=a(2)^b \\ \text{ so that is we have } \\ 4=a \text{ and } 16=4(2)^b \\ \text{ so now we only need \to solve } \\ 16=4(2)^b \\ 4=2^b \\ \text{ then we know }2^2=4 \text{ so } b=2 \\ \text{ so if } (1,4) \text{ and } (2,16) \text{ are on the graph } \\ \text{ the our function looks like } f(x)=4x^2\]
anonymous
  • anonymous
Okay. For example, how would I find a and b in the following functions: 0.000249 = a280b 0.000122 = a180b

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anonymous
  • anonymous
It is to the power of b.
myininaya
  • myininaya
\[2.49 \cdot 10^{-4}=a(280)^b \\ 1.22 \cdot 10^{-4}=a(180)^b \\ \text{ you could solve the first equation for } a \text{ then replace the } a \\ \text{ in the second equation with what you found it to be from the first } \\ \text{ this is what I did in the example \above }\] In general say we the points (c,d) and (m,n) \[d=a(c)^b \text{ and } n=a(m)^b \\\text{ solve first for } a \\ d(c)^{-b}=a \\ \text{ so now we have \in the second equation } \\ n=d(c)^{-b}(m)^b \text{ by law of exponents you have } \\ n=d(c^{-1}m)^b \\ \text{ now we are trying \to solve for } b \\ \text{ divide both sides by} d \\ \frac{n}{d}=(c^{-1}m)^b \\ \text{ take } \ln( ) \text{ of both sides } \\ \ln(\frac{n}{d})=b \ln(c^{-1}m) \\ \text{ now multiply both sides by } \frac{1}{\ln(c^{-1}m)} \\ \\ \frac{1}{\ln(c^{-1}m)}\ln(\frac{n}{d})=b \\ \text{ so we have found } b \\ \text{ now we can go back \in find } a \\ \text{ recall } a=d(c)^{-b} \\ a=d(c)^{-\frac{1}{\ln(c^{-1}m)} \ln(\frac{n}{d})}\]
myininaya
  • myininaya
so try solving your first equation for a
myininaya
  • myininaya
then pluggin that into the second equation
myininaya
  • myininaya
please let me know if you still don't follow
anonymous
  • anonymous
Will this be as accurate as linear regression, considering there are several values?
myininaya
  • myininaya
I don't know. I would have to have the set of all the points. And do a comparison with this method (whatever method this is called) versus the linear regression method Basically I would want to see if more points fall closer to the line or the the thing that we find from doing f=ax^b
myininaya
  • myininaya
by the way the value you get for a is really close to 0 using these points you chose but anyways I think I definitely would suggest writing it in scientific notation otherwise you will have a lot of zeros to write :p

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