Logarithmic functions in the biological sciences

Now, we will return to our example of a culture of bacteria that reproduces by binary fission every three hours.

In the exponential section we posed the following questions about this bacterial culture,

1. How many bacteria will be present after 51 hours if you inoculate a culture with 1 bacterium?
   
2. How many bacteria should you inoculate a culture with if there are to be 81,920 bacteria present after 42 hours?
   
   
   Now what if we pose a third question
   
3. How long would it take for an initial population of six to reach a size of 12,288 bacteria?
   
   To answer this question , you need to solve for an unknown time. How would you isolate the variable t when it is part of the exponential. To do this you need to understand logarithms and logarithmic functions.

Logarithmic and exponential functions are inverse functions

Logarithmic functions arise naturally after the study of exponential functions. Mathematically speaking, logarithmic functions and exponential functions are inverses of one another. Practically speaking, logarithms are needed to solve problems involving exponentials and vice versa. In a sense, logarithmic functions “undo” exponential functions, and exponential functions "undo" logarithmic functions.

We will use this fact to solve equations that arise in various applications. Before we get to these applications, we will develop some mathematics behind the logarithmic functions, as we did for exponential functions.

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In the next section we will provide a definition of a logarithmic function and show some examples.

Definition