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 = 73(2.6)^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 = 73(2.6)^x represents an exponential growth model, where the output increases rapidly as 'x' increases. Understanding the properties of exponential functions is crucial for rewriting them in different bases.

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 the inverse operation of the exponential function with base 'e'. When rewriting an exponential equation in terms of base 'e', the natural logarithm is used to express the exponent in a more manageable form, facilitating easier calculations and interpretations.

Change of Base Formula

The change of base formula allows us to convert logarithms from one base to another. It states that log_b(a) = log_k(a) / log_k(b) for any positive 'k'. This formula is particularly useful when rewriting an exponential equation in terms of base 'e', as it enables the transformation of logarithmic expressions, making it easier to solve for 'x' or express the equation in the desired format.