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Here is a graph that represents an exponential function with base , defined by .
A population of crickets under certain conditions can be modeled by a function defined by , where is the number of weeks since the population was first measured.
We can use the natural logarithm to solve exponential equations that are expressed with the base .
Suppose a bacteria population is modeled by the equation , where is the number of hours since the population was first measured. When will the population reach 500,000?
One way to answer this is to solve the equation , which is when .
The natural logarithm tells us the exponent to which we raise to get a given number, so . This means or about 12.4, so it takes 12.4 hours (or 12 hours and 24 minutes) for the population to reach 500,000.
We can also use a graph to solve an exponential equation. To solve , we can graph and on the same coordinate plane and find the point of intersection.
The graph shows us that the bacteria population reaches 500,000 when the input value is a little over 12, or about 12 hours after the population was first measured.