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Does a linear, exponential, or logarithmic model best fit the data in [link] ? Find the model.
x | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
y | 3.297 | 5.437 | 8.963 | 14.778 | 24.365 | 40.172 | 66.231 | 109.196 | 180.034 |
Exponential. y=2e0.5x.
While powers and logarithms of any base can be used in modeling, the two most common bases are 10 and e. In science and mathematics, the base e is often preferred. We can use laws of exponents and laws of logarithms to change any base to base e.
Given a model with the form y=abx, change it to the form y=A0ekx.
Change the function y=2.5(3.1)x so that this same function is written in the form y=A0ekx.
The formula is derived as follows
Change the function y=3(0.5)x to one having e as the base.
y=3e(ln0.5)x
Access these online resources for additional instruction and practice with exponential and logarithmic models.
Half-life formula | If A=A0ekt, k<0, the half-life is t=−ln(2)k. |
Carbon-14 dating |
t=ln(AA0)−0.000121.
A0 A is the amount of carbon-14 when the plant or animal died t is the amount of carbon-14 remaining today is the age of the fossil in years |
Doubling time formula | If A=A0ekt, k>0, the doubling time is t=ln2k |
Newton’s Law of Cooling | T(t)=Aekt+Ts, where Ts is the ambient temperature, A=T(0)−Ts, and k is the continuous rate of cooling. |
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