n Exercises 92–93, rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm and then round to three decimal places. y = 6.5(0.43)^x

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form y = a(b^x), where 'a' is a constant, 'b' is the base, and 'x' is the exponent. In this context, the function y = 6.5(0.43)^x represents an exponential decay since the base (0.43) is less than 1. Understanding the behavior of exponential functions is crucial for rewriting the equation in terms of natural logarithms.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base 'e', where 'e' is approximately equal to 2.71828. It is used to solve equations involving exponential functions by transforming them into linear forms. In this problem, converting the given exponential equation into a natural logarithm will allow us to isolate the variable 'x' and express the equation in a more manageable form.

Logarithmic Properties

Logarithmic properties are rules that govern the manipulation of logarithms, such as the product, quotient, and power rules. These properties are essential for simplifying expressions and solving equations involving logarithms. In this case, applying the power rule will help in rewriting the equation y = 6.5(0.43)^x in terms of natural logarithms, facilitating the solution process.