How do you Find the exponential growth function that models a given data set?

Ok, you can use a program such as Excel that has a function to evaluate the fitting line to a set of data but you can do it by yourself using a general function like:
#y=Ae^(kx)#
Where #A# and #k# are two constants you need to evaluate.
To evaluate the constants you use your data.
Say that you have values for #x# and corresponding values for #y#; you substitute them into your basic function and evaluate the constants.

For example:
Consider two pairs of data (I invented them):
#x=0# , #y=5#
And
#x=10# , #y=100#
We use them into our basic function to get:
#5=Ae^(0x)# where #e^0=1#
So we get #A=5#

And

#100=Ae^(10k)# but #A=5# so:
#100=5e^(10k)#
Rearranging:
#e^(10k)=100/5#
#e^(10k)=20#
Take the natural log of both sides to get rid of the exponential:
#10k=ln(20)#
#k=ln(20)/10=0.3#

So finally, the exponential function modelling our data will be:

#y=5e^(0.3x)#

Suppose that you have experimental data #(x_i,y_i)# that you believe is related by an exponential function of the form:

# y = Ce^(kt) #

where #C# and #k# are constants, and #e# is the base of Natural logarithms (Euler's Number #2.71828182...#. Here if #k># we would be modelling exponential growth (eg bacteria growth), and #k<0# would be exponential decay (eg radioactivity decay).

(NB Equality any equation of the form #y = CA^(kt) + D# can be written in the above form, so that exponential base we choose is arbitrary, and any additional constants can be incorporated into a single multiplier constant).

Then by taking Natural logarithms of both sides we get:

# lny = ln{Ce^(kt)} #
# " " = lnC + lne^(kt) #
# " " = lnC + kt #

If we write this as:

# lny = kt + lnC#

And compare with the standard equation of a straight lie@

# Y = mX +c #

Hopefully is clear that if we plot #lny# on the #y#-axis against #t# on the #x#-axis and draw a line of best fit, then #k=m# (the slope) and #lnC=c# (the #y#-intercept).

Obviously if plotting the data, does not produce a clear line of best fit then the original assumption about the data being modelled by an exponential function # y = Ce^(kt) # is incorrect.

On other note of caution, In mathematics we would generally choose base #e# because of the calculus, but often in physics base #10# is used because of scientific notation. As indicated above the base is arbitrary, choose the appropriate base and stick with it.

If you would like a solid example let me know and I will find a physics/maths exam question.