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 ln 5^2)

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form g(x) = a^x, where 'a' is a positive constant and 'x' is the exponent. The function g(x) = e^x is a specific case where the base 'e' (approximately 2.718) is used. These functions exhibit rapid growth and are fundamental in various applications, including compound interest and population growth.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions, expressed as y = log_a(x), which answers the question: 'To what exponent must the base 'a' be raised to produce x?' The natural logarithm, denoted as ln(x), uses the base 'e' and is crucial for solving equations involving exponential growth and decay, as well as in calculus.

Properties of Logarithms

The properties of logarithms, such as the product, quotient, and power rules, allow for the simplification of logarithmic expressions. For instance, log_a(b^c) = c * log_a(b) and log_a(b * c) = log_a(b) + log_a(c). Understanding these properties is essential for manipulating and evaluating logarithmic expressions, especially when combined with exponential functions.