Determine the exponential regression equation for the following set of data
X: 0, 5, 10, 15, 20, 25
Y: 30.9, 14.7, 7.5, 4.2, 2.3, 1
I know the exponential regression equation is;
Using your equation, we simply have to isolate x.
`y = 29.8969(0.8753)^x`
It's easy to find y as you simply have to substitute your given x to the equation, and then simplify the equation. The same is true when we need to find x; however, the calculation is more tedious. We know that y = 7:
`7 = 29.8968(0.8753)^x`
We first divide by 29.8968. This will help us isolate x.
`7/(29.8968) = 0.8753^x`
Now, to 'bring down' x (or isolate x) we have to take the logarithm of both sides (it doesn't matter which logarithm base we use as it will give the same answer; we'll use base 10 here for no particular reason.) We are getting logarithms of both sides because one property of logarithms is that the log of something raised to an exponent is equal to the log of that something multiplied by the magnitude of the exponent: `log a^b = b log a`
`log[7/(29.8968)] = log(0.8753)^x = x log(0.8753)`
Now, we can easily isolate x:
`x = (log[7/(29.8968)])/(log(0.8753)) approx 10.9`
which gives you the x value you calculated :)