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,
Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time. We use the command "LnReg" on a graphing utility to fit a logarithmic function to a set of data points. This returns an equation of the form,
Due to advances in medicine and higher standards of living, life expectancy has been increasing in most developed countries since the beginning of the 20th century.
The table below shows the average life expectancies, in years, of Americans from 1900–2010.
Use the "LnReg" command from the STAT then CALC menu to obtain the logarithmic model,
Next, graph the model in the same window as the scatterplot to verify it is a good fit.
If life expectancy continues to increase at this pace, the average life expectancy of an American will be 79.1 by the year 2030.
Sales of a video game released in the year 2000 took off at first, but then steadily slowed as time moved on. The table below shows the number of games sold, in thousands, from the years 2000–2010.
|Number Sold (thousands)||142||149||154||155||159||161|
|Number Sold (thousands)||163||164||164||166||167||—|
a. Let x represent time in years starting with x = 1 for the year 2000. Let y represent the number of games sold in thousands. Use logarithmic regression to fit a model to these data.
b. If games continue to sell at this rate, how many games will sell in 2015? Round to the nearest thousand.