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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.

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Expressing an exponential model in base e

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.

  1. Rewrite y=abx as y=aeln(bx).
  2. Use the power rule of logarithms to rewrite y as y=aexln(b)=aeln(b)x.
  3. Note that a=A0 and k=ln(b) in the equation y=A0ekx.

Changing to base e

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

y=2.5(3.1)x=2.5eln(3.1x)Insert exponential and its inverse.=2.5exln3.1Laws of logs.=2.5e(ln3.1)xCommutative law of multiplication
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Change the function y=3(0.5)x to one having e as the base.

y=3e(ln0.5)x

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Key equations

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.

Key concepts

  • The basic exponential function is f(x)=abx. If b>1, we have exponential growth; if 0<b<1, we have exponential decay.
  • We can also write this formula in terms of continuous growth as A=A0ekx, where A0 is the starting value. If A0 is positive, then we have exponential growth when k>0 and exponential decay when k<0. See [link] .
  • In general, we solve problems involving exponential growth or decay in two steps. First, we set up a model and use the model to find the parameters. Then we use the formula with these parameters to predict growth and decay. See [link] .
  • We can find the age, t, of an organic artifact by measuring the amount, k, of carbon-14 remaining in the artifact and using the formula t=ln(k)0.000121 to solve for t. See [link] .
  • Given a substance’s doubling time or half-time, we can find a function that represents its exponential growth or decay. See [link] .
  • We can use Newton’s Law of Cooling to find how long it will take for a cooling object to reach a desired temperature, or to find what temperature an object will be after a given time. See [link] .
  • We can use logistic growth functions to model real-world situations where the rate of growth changes over time, such as population growth, spread of disease, and spread of rumors. See [link] .
  • We can use real-world data gathered over time to observe trends. Knowledge of linear, exponential, logarithmic, and logistic graphs help us to develop models that best fit our data. See [link] .
  • Any exponential function with the form y=abx can be rewritten as an equivalent exponential function with the form y=A0ekx where k=lnb. See [link] .
Practice Key Terms 6

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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