In Exercises 19–29, evaluate each expression without using a calculator. If evaluation is not possible, state the reason. ln e^5

Key Concepts

Natural Logarithm (ln)

The natural logarithm, denoted as 'ln', is the logarithm to the base 'e', where 'e' is an irrational constant approximately equal to 2.71828. It is used to solve equations involving exponential growth or decay and has unique properties that simplify calculations, particularly when dealing with exponential functions.

Properties of Logarithms

Logarithms have several key properties that facilitate their manipulation. One important property is that ln(a^b) = b * ln(a). This means that the logarithm of a number raised to a power can be simplified by bringing the exponent in front as a multiplier, which is crucial for evaluating expressions involving exponents.

Exponential Function

An exponential function is a mathematical function of the form f(x) = a * e^(bx), where 'a' and 'b' are constants. The function grows rapidly as 'x' increases, and its inverse is the natural logarithm. Understanding the relationship between exponential functions and their logarithmic counterparts is essential for evaluating expressions like ln(e^5).