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)

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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.

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 $\ln \ln 25$.

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".

Apply this property to $g(\ln \ln 25)$: since $g(x) = e^x$, then $g(\ln \ln 25) = e^{\ln \ln 25} = \ln 25$.

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.

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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).

Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given g(x) = ex, find g(ln ln 52)