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 5²)

Verified step by step guidance

1. Start by understanding the given function: \(g(x) = e^x\). We need to find \(g(\ln \ln 5^2)\), which means we will substitute \(x\) with \(\ln \ln 5^2\) in the function.
2. Rewrite the expression inside the logarithms step-by-step. First, simplify the exponent inside the innermost expression: \$5^2$ becomes \(25\), so the expression becomes \(\ln \ln 25\).
3. Recall the property of logarithms: \(g(\ln a) = e^{\ln a} = a\). This means that applying \(e^{(\ln (\text{something}))}\) simplifies to just that "something".
4. Apply this property to \(g(\ln \ln 25)\): since \(g(x) = e^x\), then \(g(\ln \ln 25) = e^{\ln \ln 25} = \ln 25\).
5. Thus, the original expression simplifies to \(\ln 25\). You can leave the answer in this form or further simplify \(\ln 25\) if needed using logarithm properties.

Key Concepts

Exponential Functions

Exponential functions have the form f(x) = a^x, where the variable is in the exponent. The function g(x) = e^x uses the constant e (~2.718), which is the base of natural logarithms. Understanding how to evaluate and manipulate these functions is essential for solving expressions involving e raised to a power.

Logarithmic Functions and Natural Logarithm

Logarithmic functions are the inverses of exponential functions. The natural logarithm, denoted ln(x), is the logarithm with base e. It satisfies the property ln(e^x) = x, which is key to simplifying expressions where logarithms and exponentials are nested.

Properties of Logarithms and Exponents

Key properties include ln(a^b) = b ln(a) and e^{ln(x)} = x for x > 0. These allow simplification of complex expressions by converting powers inside logarithms or exponents into products or by canceling inverse operations. Applying these properties step-by-step helps evaluate nested expressions like g(ln ln 5^2).