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When you graph an exponential function like \( y = (1.085)^{x} \), you are looking at a curve that shows a specific kind of growth. This function starts at the point \((0,1)\), meaning when \( x = 0 \), \( y \) is always 1, because any number raised to the power of zero is 1. As \( x \) increases, the value of \( y \) increases exponentially. This results in a rapid upward curve after a certain point, especially as the base of the exponent is slightly greater than 1.

To sketch this graph, start placing points using various values of \( x \), such as \( x = 0, 10, 20 \), and even the given problems like \( x = 40 \). As you plot these, you'll notice how quickly the values rise. The graph will steadily slope upward and never touch the x-axis, indicating it's always positive.

This quick rise is characteristic of exponential functions, where even small increases in \( x \) lead to large increases in \( y \). Understanding this helps you predict the behavior of exponential growth functions in real-world situations, such as population growth or compound interest.

Exponentiation involves raising a base number to the power of an exponent, like in \((1.085)^{x}\). Here, 1.085 is the base, and \( x \) is the exponent. The exponent tells you how many times to multiply the base number by itself.

The base value is crucial in determining the function’s nature:

• If the base is greater than 1, the function will depict exponential growth, leading to a rapidly increasing line.
• If the base is less than 1, the function will show exponential decay, where the curve will move downward as \( x \) increases.

In the original exercise, the base is 1.085, indicating growth. By understanding this exponentiation concept, you can calculate outcomes for different values of \( x \) by simply raising the base to that power. That’s how you determine \( y \) for any \( x \) or solve for \( x \), given a \( y \) value.

Logarithms are the inverse operations of exponentiation, which means they answer the question: "To what exponent must we raise the base to get a specific number?" When you encounter a situation like solving for \( x \) in \((1.085)^{x} = 2\), you need logarithms to find \( x \).

Using the logarithm function, you can rearrange the equation:

• Take the logarithm of both sides: \( \log((1.085)^{x}) = \log(2) \).
• Use the logarithm power rule to simplify: \( x \cdot \log(1.085) = \log(2) \).
• Solve for \( x \): \( x = \frac{\log(2)}{\log(1.085)} \).

This is how you estimate \( x \) when the resulting \( y \) value is known. Logarithms help you work backward from an exponentiated number to find out what the exponent was. This technique is particularly handy not just in math but also in calculating time and growth in various fields, such as science and finance.