Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given g(x) = e^x, find g(ln 4)

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form g(x) = a^x, where 'a' is a positive constant. The function g(x) = e^x is a specific case where the base 'e' is the natural logarithm base, approximately equal to 2.718. These functions exhibit rapid growth and are characterized by their unique property that the rate of change is proportional to the function's value.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base 'e'. It is the inverse operation of the exponential function with base 'e'. This means that if y = ln(x), then e^y = x. Understanding this relationship is crucial for evaluating expressions involving natural logarithms and exponential functions.

Properties of Logarithms

Logarithms have several key properties that simplify calculations, including the product, quotient, and power rules. For instance, ln(a*b) = ln(a) + ln(b) and ln(a/b) = ln(a) - ln(b). Additionally, ln(e^x) = x, which is essential for evaluating expressions where the natural logarithm and exponential functions are involved, allowing for straightforward simplification.