If a bacteria population starts with 100 bacteria and...

If a bacteria population starts with 100 bacteria and doubles every three hours, then the number of bacteria after $ t $ hours is $ n = f(t) = 100 \cdot 2^\frac{t}{3} $.

(a) Find the inverse of this function and explain its meaning.
(b) When will the population reach 50,000?

Key Concepts

Exponential Growth Modeling

Exponential growth modeling describes situations where a quantity increases by a constant percentage over time, leading to a rapid increase in size. This concept is widely applicable in biology, finance, and physics, and understanding it is crucial for predicting future values and determining growth-related time points.

Logarithms

Logarithms are the inverse operations of exponentiation. They are instrumental in solving for the exponent in equations where the variable appears in the exponent. When finding the inverse of an exponential function, logarithms allow us to 'undo' the exponential process and isolate the variable of interest.

Exponential Functions

Exponential functions model processes that increase or decrease at rates proportional to their current value. They are used in various fields to represent growth phenomena, such as population dynamics, where the change over time is multiplicative rather than additive.

Inverse Functions

An inverse function reverses the roles of inputs and outputs, effectively solving for the original independent variable. In the context of exponential functions, finding the inverse involves determining the time or input required to obtain a specific output, effectively switching the dependent and independent variables.

Transcript

Please give Ace some feedback