The natural logarithm is the inverse function of which exponential function?

• A. e^(-x)
• B. 10^x
• C. 2^x
• D. e^x

Answer: (D)

The natural logarithm is the inverse of the exponential with base e. That means ln(e^x) = x for all x, and e^{ln x} = x for x > 0. The exponential with base e maps all real numbers to positive outputs, and the natural log takes positive inputs back to real numbers, so they undo each other. The other exponentials would pair with logarithms of their respective bases (log base 10 for 10^x, log base 2 for 2^x), and the inverse of e^{-x} is -ln x, not ln x. So the base-e exponential, e^x, is the one whose inverse is the natural logarithm.