4.7 Exponential and logarithmic models  (Page 7/16)

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 $e.$

Key equations

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 $A=A_0{e}^{kx},$ 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 $t=\frac{\ln \left(k\right)}{-0.000121}$ 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 $y=A_0{e}^{kx}$ where $k=\ln b.$ See [link] .