In Exercises 81–100, evaluate or simplify each expression without using a calculator. e^(ln 5x^2)

Key Concepts

Exponential and Logarithmic Functions

Exponential functions involve a constant base raised to a variable exponent, while logarithmic functions are the inverse operations of exponentials. The natural logarithm, denoted as 'ln', uses the base 'e' (approximately 2.718). Understanding the relationship between these functions is crucial for simplifying expressions involving 'e' and 'ln'.

Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as the power rule, which states that ln(a^b) = b * ln(a). This property allows us to manipulate logarithmic expressions effectively, making it easier to evaluate or simplify complex expressions involving logarithms and exponentials.

Inverse Functions

Exponential and logarithmic functions are inverses of each other, meaning that applying one function followed by the other returns the original value. For example, e^(ln(x)) = x. This property is essential for simplifying expressions like e^(ln(5x^2)), as it allows us to directly evaluate the expression without complex calculations.