Assignment 19

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course mth 163

7/6 5pm

019. `query 19*********************************************

Question: `qexplain the steps in fitting an exponential function to data

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Your solution:

You would take the data you have in the y column of your table and get the log(y) of those numbers. Then do a linear regression with the data in the table of x and log(y) to get that data in the form y = mx+b, but it will really mean log(y) = mx+b.

Then y = 10^log(y) = 10^(mx) * 10^(x) this will give y = m^x *b.

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Given Solution:

`a** If you have two points you can solve the simultaneous equations:

• Substitute the coordinates into the form y = A b^x and solve the two resulting equations for A and b.

• You could alternatively use the form y = A * 2^(k x) or y = A * e^(k x), in which case you would solve for A and k.

If you have a more extensive data set you can use transformations.

For exponential data you plot log(y) vs. x. If the graph is well approximated by a straight line then you get an exponential function.

Then since the graph is a straight line, you can find its equation using using either slope and vertical intercept, or two points on the line.

If the slope of a y vs. x graph is m and the vertical intercept is b then the function is y = m x + b.

However in this case the graph is not of y vs. x, but of log(y) vs. x.

So if the slope of your graph is m and the y intercept is b, the function is log(y) = m x + b.

This equation needs to be solved for y:

You invert the transformation using the inverse function 10^x, obtaining 10^log(y) = 10^(mx+b).

10^log(y) = y, by the definition of the logarithm, and

10^(mx + b) = 10^(mx) * 10^b, by the laws of exponents.

Thus

• y = 10^(mx) * 10^b,

where m and b are just the numbers (slope and vertical intercept) that you determined from your graph.

Note that if a power function fits the data then log y vs. log x will give a straight line so that log y = m log x + b. In this

case our solution will be y = 10^b * x^m, a power function rather than an exponential function. **

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