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 1/e)

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 Euler's number, approximately equal to 2.71828. 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 function of the exponential function g(x) = e^x. This means that if y = ln(x), then e^y = x. Understanding the properties of logarithms, such as ln(a/b) = ln(a) - ln(b) and ln(a^b) = b*ln(a), is essential for manipulating expressions involving logarithms.

Inverse Functions

Inverse functions are pairs of functions that reverse the effect of each other. For example, if f(x) is an exponential function, its inverse is the logarithmic function. In the context of the question, evaluating g(ln(1/e)) involves recognizing that ln(1/e) simplifies to -1, allowing us to find g(-1) = e^(-1) = 1/e, demonstrating the relationship between exponential and logarithmic functions.