Earlier, we studied exponential relationships where the input values are the exponent in the function. Sometimes we want to express an exponential relationship where the values we want to find, the outputs, are the exponents. A logarithmic function, or a function of the form , where is a constant and is a positive number, can help us do that.

For example: Suppose the population of a town starts at one thousand and doubles every decade since first measured. We can write  (or ) to represent the population, in thousands, after decades.

But if we want to know how long, in decades, it would take to reach certain population sizes, in thousands, we can write a logarithmic function . In this function, the input is , population in thousands, and the output is , time in decades. Here is a graph representing that function.

We can use the graph to estimate the answer to a question such as, “How many decades will it take for the population to reach a million?” by finding the value 1,000 (1,000 thousands is 1 million) on the horizontal axis and following that up to see that the graph has a point near . This means that the population will reach 1 million after about 10 decades. We can check the answer by finding that .Suppose the population of that town expands by a factor of 10 every decade instead of by a factor of 2. The function representing the time it takes to reach a certain population, in thousands, would be .

From the graph, we can see that it takes only 3 decades to reach 1,000 thousands, because the point appears to be on the graph (and ).