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)

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Recall the given function: \(g(x) = e^x\).

Identify the input to the function: \(\ln \left( \frac{1}{e} \right)\).

Use the logarithmic property that \(\ln \left( \frac{1}{a} \right) = -\ln a\) to rewrite the input as \(-\ln e\).

Since \(\ln e = 1\), simplify the input to \(-1\).

Evaluate \(g(-1)\) by substituting into the function: \(g(-1) = e^{-1}\).

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Exponential Functions

An exponential function has the form f(x) = a^x, where the base a is a positive constant. The function g(x) = e^x uses the natural base e (~2.718), which is fundamental in continuous growth and decay models. Understanding how to evaluate e raised to various powers is essential for solving the given problem.

Logarithmic Functions and Natural Logarithm

A logarithmic function is the inverse of an exponential function. The natural logarithm, denoted ln(x), is the logarithm with base e. It satisfies the property ln(e^x) = x, which helps simplify expressions involving e and ln, such as ln(1/e).

Properties of Logarithms and Exponents

Key properties include ln(a/b) = ln(a) - ln(b), ln(e) = 1, and e^{ln(x)} = x for x > 0. These properties allow rewriting and simplifying expressions like ln(1/e) into manageable forms, enabling direct evaluation of functions like g(ln(1/e)).

Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given g(x) = ex, find g(ln 1/e)