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Natural logarithms are a fundamental part of mathematics used particularly in cases involving exponential functions. The natural logarithm, written as \( \ln(x) \), is the inverse function of an exponential \( e^x \). This means when you have \( e^x = y \), you can find \( x \) by calculating \( \ln(y) \).

• Natural logs are base \( e \), where \( e \approx 2.71828 \), a mathematical constant.
• This operation helps when working with exponential decay or growth scenarios.

In the given problem, the function describing the number of cars \( N \) is dependent on an exponential decay term \( e^{-2t} \). To solve for \( t \), the natural logarithm is employed. By taking the natural log of both sides of the equation \( e^{-2t} = 0.1 \), we effectively "bring down" the exponent, making it easier to solve for \( t \). This simplifies the expression to \( -2t = \ln(0.1) \), allowing straightforward solving for \( t \).
Natural logarithms simplify complex equations, making it possible to solve them using arithmetic. They are very useful in contexts where variables are exponents.

Solving equations, especially those involving exponential functions, often requires breaking them down into simpler steps. In the exercise, we start with the equation \( \frac{100,000}{1+10 e^{-2 t}} = 50,000 \) and manipulate it to isolate \( t \). Here's the step-by-step approach:

• First, eliminate fractions by multiplying both sides by \( 1 + 10e^{-2t} \).
• Next, distribute and simplify the equation to extract the exponential term.
• After isolating \( e^{-2t} \), logarithmic operations can be applied to both sides.

This method demonstrates how we isolate variables, especially those within exponents, to simplify and solve for desired values. Each step, such as multiplying through to eliminate denominators, is crucial as it leads toward the ultimate goal of simplifying the complexity of the equation. In general, solving equations involves a strategic process where simplifying and rearranging terms leads to an understandable solution.

Function analysis is an essential part of understanding how changes in variables affect the whole equation. The original function \( N = \frac{100,000}{1+10 e^{-2 t}} \) describes how many Honda Accords are on the road depending on weeks passed. By analyzing this function, we can predict future behavior.

• The function shows a type of growth limited by the maximum value of 100,000.
• This is indicative of a saturation point or asymptote, typical in logistic growth models.

As time \( t \) increases, the growth of \( N \) slows, reflecting real-world models where resources or populations can't grow indefinitely. Understanding such features allows us to grasp the relationships between time, growth, and limits without complex calculations.
Function analysis also involves checking the behavior at extreme values. As \( t \rightarrow \infty \), \( e^{-2t} \rightarrow 0 \), leading \( N \) to approach its maximum value of 100,000. These insights help in anticipating the long-term expectations of the model, offering a comprehensive understanding of the situation.