3.2.2—Exponential and ln

e^{ln x}=x for x>0 and ln(e^x)=x for real x. Exponential and logarithmic graphs reflect in y=x, with domains and ranges exchanged.

When solving, isolate the exponential or logarithm, then apply the inverse and preserve positivity. For models, identify whether the exponent is linear in the independent variable.

e^{2x−1}=7 gives x=(1+ln7)/2; a logarithm of a non-positive quantity is not defined over the reals.

ln(e^{x}) and e^{ln x} have different domain conditions even though both simplify algebraically.