Final Answer:
The f(x) population exceeds the g(x) population in month x = ln(4)/ln(2), which is approximately x ≈ 2. This means that the f(x) population surpasses the g(x) population in the second month.
Step-by-step explanation:
Bacterial populations can be modeled using exponential growth functions, where f(x) and g(x) represent the populations of two bacterial groups over time. The point at which f(x) surpasses g(x) can be determined by finding the x-value when f(x) becomes greater than g(x). In mathematical terms, this occurs when f(x) > g(x).
The given information indicates exponential growth, and comparing the two functions involves finding the solution to the inequality f(x) > g(x). Let's consider the functions f(x) = 2^x and g(x) = 4^x. To find when f(x) > g(x), we set 2^x > 4^x and simplify. Taking the logarithm of both sides allows us to solve for x:
x . ln(2) > x . ln(4)
Canceling out x from both sides, we get:
ln(2) > ln(4)
Now, using the property of logarithms that \(\ln(a^b) = b \cdot \ln(a)\), we simplify further:
1 > 2 . ln(2)/ln(2)
Finally, dividing both sides by 2:
1/2 > \ln(2)/ln(2)
Therefore, the solution is x = ln(4)/ln(2), which is approximately 2.
In conclusion, the f(x) population surpasses the g(x) population in the second month, as x = ln(4)/ln(2). This analysis provides a mathematical understanding of when the two bacterial populations diverge in growth.