How is the Logarithm derived from the exponential function? (aren't they inverses?)

I've been learning logs in school, and my teacher, friend, and I are stumped on something. How does one derive the logarithmic function from the exponential function? My friend thinks Tayler Series are the trick. Is he right? Is there a a better/simpler/more elegant way? Also, do calculators use taylor series to do logs? Thanks for the help

I've been learning logs in school, and my teacher, friend, and I are stumped on something. How does one derive the logarithmic function from the exponential function? My friend thinks Tayler Series are the trick. Is he right? Is there a a better/simpler/more elegant way? Also, do calculators use taylor series to do logs? Thanks for the help