In Exercises 39–40, graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x) = ln x and g(x) = - ln (2x)

Logarithmic functions, such as f(x) = ln(x), are the inverses of exponential functions. They are defined for positive real numbers and have a vertical asymptote at x = 0. Understanding their properties, including their domain, range, and behavior as x approaches the asymptote, is crucial for graphing and analyzing transformations.

Transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For example, g(x) = -ln(2x) represents a vertical reflection of f(x) = ln(x) and a horizontal compression. Recognizing how these transformations affect the graph helps in visualizing and deriving the new function's characteristics from the original.

Asymptotes are lines that a graph approaches but never touches. For logarithmic functions, vertical asymptotes occur where the function is undefined, such as at x = 0 for f(x) = ln(x). Identifying asymptotes is essential for understanding the behavior of the function at its boundaries and for determining the domain and range of the functions involved.