What is the exponential function y=Ce^kt that passes through two given points?

Question

Find the exponential function y = Ce^kt that passes through the two given points.

Given points: (4,5) and (3,1/2)

Answer 1

So this is basically an algebra question. First, you will plug in 5 for t and 4 for y.

You can solve for either C or k as a function of the other one.

Then, you plug in C into the original equation, but now you plug in 3 for t and 0.5 for y. At this point, all that is unknown is k.

k, as you know, can be tricky to solve for. As such, you have to remember, what is the reverse of an exponential.

That's right, it is ln (natural log). If you take the natural log of both sides, the k is now solvable through simple division.

Now, C is still a function of k. You must plug in the newfound value of k into THAT function to get the numerical value of C.

You can test your answer by plugging in the original points of 5,4 and 3,0.5 to see if you did it correctly.

Answer 2

5 = C*exp(4k)

1/2 = C*exp(3k)

Multiplies second equation by 10:

5 = 10*C*exp(3k)

sets the equations equal, since they both equal 5:

10*C*exp(3k) = C*exp(4k)

10*exp(3k) = exp(4k) : divides both sides by C, which can't be zero, otherwise it would not

be an exponential function (exponential would cancel out)

Takes log of both sides:

ln[ 10 * exp(3k)] = 4k

ln 10 + 3k = 4k

ln 10 = k

So then

y = C * exp(ln 10 * t)

= C * [exp(ln 10) ] ^t <--- law of exponents

= C * 10^t

5 = C* 10^4

5 / 10000 = C

1/2000 = C

Exponential function is:

y = (1/2000) * 10^t