Final answer:
The equation y' = ln(y) is used in exponential regression. This approach is used to linearize data that represent processes growing or decaying at a proportional rate, enabling us to apply linear regression to the transformed data.
Step-by-step explanation:
You would use the equation y' = ln(y) in the context of exponential regression. This is because the natural logarithm function, represented as ln, is the inverse of the exponential function. When we have data that grow (or decay) at a rate proportional to its current size, it typically fits an exponential model. In cases like these, taking the natural logarithm of the data can linearize the relationship, allowing us to apply linear regression techniques on the transformed data to find the best fit. This process is useful in situations such as tracking the growth of a population or the decay of a radioactive substance over time. It's often employed in scientific fields such as biology for modeling population dynamics or in chemistry for analyzing reaction kinetics.
For instance, if a plot of reactant concentration versus time is not linear, but plotting the natural logarithm of reactant concentration versus time is linear, then the reaction follows first-order kinetics, which requires exponential modeling. Similarly, when dealing with compound interest or physical phenomena following doubling periods (for example, population growth), exponential regression is used to model the underlying processes accurately.