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Understanding the exponential growth equation is crucial for solving problems involving the rapid increase of quantities, such as populations or investments. Exponential growth is represented by the equation \[ N(t) = N_0e^{rt} \]where

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• \t\t\( N(t) \) is the amount at time \( t \).\t
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• \t\t\( N_0 \) is the initial amount, or starting value.\t
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• \t\t\( r \) is the growth rate, which indicates how quickly the amount is increasing over time.\t
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• \t\t\( e \) is the base of the natural logarithm, an irrational constant approximately equal to 2.71828.\t

The power to which \( e \) is raised, \( rt \), is the product of the growth rate and time, indicating that growth compounds exponentially. This means that as time progresses, the rate of increase itself becomes larger, leading to a characteristic steep 'curve up' in graphical representations. It's important to note that when the quantity doubles over regular intervals, as in the given exercise, exponential growth is at play.

When solving exponential equations, the basic approach is to isolate the term with the exponent and then use logarithms to solve for the variable. As exemplified in the bacteria population problem, we have to create two equations based on the given information and then find the value of the growth rate \( r \). The step-by-step process often involves the following steps:

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• Setting up the exponential growth equation with the initial conditions.
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• Creating additional equations based on given data points or conditions.
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• Isolating the exponential term which often includes the unknown variable.
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• Taking the natural logarithm of both sides to convert the equation from an exponential form to a linear form, making it easier to solve.
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• Finding the numerical value of the variable using algebra.

Using these steps, we can express relationships that initially seem complicated by exponential terms in a more straightforward, solvable manner. This process is fundamental for understanding a wide range of problems involving exponential growth, including those in biology, finance, and physics.

The natural logarithm, denoted as \( \ln \), is an operation that is crucial for unraveling equations involving exponential growth. It is the inverse of raising the number \( e \) to a power, answering the question: 'To what power must \( e \) be raised to produce a given number?' For example, if you have \( e^x = y \), then \( \ln(y) = x \).

This concept plays a pivotal role when solving for variables in exponential equations because it allows us to turn the multiplication of the exponent into addition or subtraction, a simpler operation to handle algebraically. As seen in the exercise solution, by taking the natural logarithm of both sides of the equation \( 2 = e^{9r} \), we convert the exponential equation into a linear one, allowing us to solve for the growth rate \( r \) directly. Understanding how to use the natural logarithm is essential for students tackling subjects that involve any sort of exponential changes, and it's a key element of logarithmic properties that extend into many areas of mathematics and science.