How do you find the exponential model #y=ae^(bx)# that goes through the points (0,1) and (3,10)?

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Answer 1 · 2017-03-01T01:00:15

#y= 10^(x/3)#

We know two #(x, y)# points, so we have enough information to write a system of equations and solve for #a# and #b#.

Equation 1:

#1 = ae^(0b)#

Equation 2:

#10 = ae^(3b)#

The first equation can be simplified to #a = 1#, because #0(a)# will always equal #0# and any real number #x# has the property such that #x^0 = 1#.

Solve for #b# now.

#10 = 1(e^(3b))#

#10 = e^(3b)#

#ln10 = ln(e^(3b))#

#ln10 = 3blne#

#3b = ln10#

#b = 1/3ln10#

The function therefore has equation #y = e^(1/3ln10x)#. Now let's look at simplifying the function.

Use the logarithm property #x^a = e^(alnx)#, to arrive at the result

#y = (10^(1/3))^x = 10^(x/3)#

Hopefully this helps!