Solving exponential equations

Question

The exponential function y(x)=Ce^alpha x satisfies the condition y(0)=6 and y(1)=5.

Find C and alpha.

 

Find y(2) and give the exact answer.

Answer 1

I think that you mean y(x) = Ce^αx.

 

Since y(0) = 6, Ce⁰= 6.  So, C = 6

 

Therefore, y(x) = 6e^αx

 

Since y(1) = 5, we have 6e^α = 5

 

                                    e^α = 5/6   So, α = ln(5/6) ≈ -0.18

 

Thus, y(x) ≈ 6e^-0.18x

 

 

Answer 2

Assuming that the given is y(x) = Ce^{(a^x)} is a correct representation of the given, and using a to represent alpha:

When x = 0, then the exponent a^0 is 1, so 6 = Ce; therefore C = 6/e

When x = 1, then 5 = Ce¹ so 5 = (6/e)e^a, or 5/6 = e ^{a - 1}

Taking natural logarithms: ln (5/6) = a - 1  so a = 1 + ln (5/6).