4.8 Fitting exponential models to data  (Page 3/12)

Building a logarithmic model from data

Just as with exponential functions, there are many real-world applications for logarithmic functions: intensity of sound, pH levels of solutions, yields of chemical reactions, production of goods, and growth of infants. As with exponential models, data modeled by logarithmic functions are either always increasing or always decreasing as time moves forward. Again, it is the way they increase or decrease that helps us determine whether a logarithmic model is best.

Recall that logarithmic functions increase or decrease rapidly at first, but then steadily slow as time moves on. By reflecting on the characteristics we’ve already learned about this function, we can better analyze real world situations that reflect this type of growth or decay. When performing logarithmic regression analysis , we use the form of the logarithmic function most commonly used on graphing utilities, $y=a+b\ln \left(x\right).$ For this function

• All input values, $x,$ must be greater than zero.
• The point $\left(1,a\right)$ is on the graph of the model.
• If $b>0,$ the model is increasing. Growth increases rapidly at first and then steadily slows over time.
• If $b<0,$ the model is decreasing. Decay occurs rapidly at first and then steadily slows over time.