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)

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

We need to find \(g(\ln 4)\), which means substituting \(x\) with \(\ln 4\) in the function.

Substitute to get: \(g(\ln 4) = e^{\ln 4}\).

Use the property of exponents and logarithms that states \(e^{\ln a} = a\) for any positive \(a\).

Therefore, \(g(\ln 4) = 4\) by applying this property.

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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 exponential functions at given inputs is essential.

Logarithmic Functions

A logarithmic function is the inverse of an exponential function, defined as log_a(x), which answers the question: to what power must the base a be raised to get x? The natural logarithm ln(x) uses base e and is key to solving equations involving e^x.

Inverse Properties of Exponential and Logarithmic Functions

Exponential and logarithmic functions are inverses, so applying one after the other cancels out, e.g., e^(ln x) = x for x > 0. This property allows simplification of expressions like g(ln 4) = e^(ln 4) = 4, making evaluation straightforward.

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