The inverse of the natural logarithm function ln is the exponential function with base eee, written as exe^xex.

Quick Scoop

  • If y=ln⁡(x)y=\ln(x)y=ln(x), then x=eyx=e^yx=ey.
  • So the inverse function of f(x)=ln⁡(x)f(x)=\ln(x)f(x)=ln(x) (with x>0x>0x>0) is f−1(x)=exf^{-1}(x)=e^xf−1(x)=ex.
  • They “undo” each other:
    • ln⁡(ex)=x\ln(e^x)=xln(ex)=x for all real xxx.
* eln⁡(x)=xe^{\ln(x)}=xeln(x)=x for x>0x>0x>0.

Tiny story to remember it

Think of exe^xex as a machine that takes time in and tells you how much something has grown by that time.

The natural log ln⁡(x)\ln(x)ln(x) is the opposite machine: it takes the growth and tells you how long it would have taken, so it must be the inverse of exe^xex.

In short: what is the inverse of ln? It’s the exponential function exe^xex.

TL;DR: Inverse of ln⁡(x)\ln(x)ln(x) is exe^xex.

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