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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 = 2 e 0.5 x .

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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 = a b x , change it to the form y = A 0 e k x .

  1. Rewrite y = a b x as y = a e ln ( b x ) .
  2. Use the power rule of logarithms to rewrite y as y = a e x ln ( b ) = a e ln ( b ) x .
  3. Note that a = A 0 and k = ln ( b ) in the equation y = A 0 e k x .

Changing to base e

Change the function y = 2.5 ( 3.1 ) x so that this same function is written in the form y = A 0 e k x .

The formula is derived as follows

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

y = 3 e ( ln 0.5 ) x

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Key equations

Half-life formula If   A = A 0 e k t , k < 0 , the half-life is   t = ln ( 2 ) k .
Carbon-14 dating t = ln ( A A 0 ) 0.000121 .
A 0   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 = A 0 e k t , k > 0 , the doubling time is   t = ln 2 k
Newton’s Law of Cooling T ( t ) = A e k t + T s , where   T s   is the ambient temperature,   A = T ( 0 ) T s , and   k   is the continuous rate of cooling.

Key concepts

  • The basic exponential function is f ( x ) = a b x . 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 = A 0 e k x , where A 0 is the starting value. If A 0 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 = a b x can be rewritten as an equivalent exponential function with the form y = A 0 e k x where k = ln b . See [link] .

Questions & Answers

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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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