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Exponential functions are mathematical expressions in which a variable appears in the exponent. In simpler terms, it means that the output of the function increases or decreases at a constant multiplicative rate.
Here's an example: \( f(x) = a \times b^x \). Here, \(a\) is a constant, called the coefficient, and \(b\) is the base of the exponential function. If \(b > 1\), the function represents exponential growth. If \(0 < b < 1\), then it represents exponential decay.
Exponential functions are very useful in various fields such as biology, finance, physics, and more.
In our exercise, we have an exponential equation, \( 6(1.024)^{x-1900} = 9 \). The base of our exponential function is \( 1.024 \), which indicates a growth factor.

The natural logarithm, often represented as \( \ln \), is a special logarithm with the base \( e \) (where \( e \) is approximately 2.71828). It is a very powerful tool in mathematics, especially when dealing with exponential functions.
The natural logarithm of a number n is the power to which e must be raised to equal n. In other words, if \( e^y = n \), then \( y = \ln(n) \).
In our exercise, to solve the equation for x, we use the natural logarithm to 'bring down' the exponent, making it easier to isolate \( x \).
By applying the natural logarithm to both sides of \( (1.024)^{x-1900} = 1.5 \), we get \( \ln{(1.024)^{x-1900}} = \ln{1.5} \).

Logarithmic properties can simplify complex exponential equations. Here are a few key properties:

• Power Rule: \( \ln(a^b) = b \ln(a) \)
• Product Rule: \( \ln(ab) = \ln(a) + \ln(b) \)
• Quotient Rule: \( \ln\left( \frac{a}{b} \right) = \ln(a) - \ln(b) \)

These properties help us manipulate and solve logarithmic equations with ease.
In our exercise, we use the Power Rule to simplify the equation: \( \ln{(1.024)^{x-1900}} = \ln{1.5} \). By applying the Power Rule, this becomes \( (x-1900) \ln{1.024} = \ln{1.5} \). Finally, we isolate \( x \) and solve it: \( x = 1900 + \frac{\ln{1.5}}{\ln{1.024}} \).
This demonstrates the power and utility of understanding logarithmic properties in solving exponential equations.