Exponential Function: Properties and Definitions

This lecture covers the properties of the natural logarithm, demonstrating its injectivity and surjectivity. It defines the exponential function as the inverse of the natural logarithm and explores its key properties, such as exp(-x) = 1/exp(x) and exp(x+y) = exp(x) * exp(y). The lecture also discusses the continuity and monotonicity of the exponential function, proving that exp(x) tends to infinity as x approaches infinity and to 0 as x approaches negative infinity. Additionally, it examines the relationship between the exponential and logarithmic functions, showcasing how exp(ln(b)) = b and exp(ln(b)^2) = b^2 for any positive real number b.