2.2.2—Exponential and ln

For a>0,a≠1, y=a^x has inverse x=log_a y. The natural logarithm ln x is log_e x, and ln(e^x)=x for real x while e^{ln x}=x only for x>0.

Use the inverse relation to solve for a variable, and preserve positivity when taking logarithms. Exponential growth has constant proportional rate; logarithmic growth has decreasing gradient.

3e^{2x}=12 gives e^{2x}=4 and x=½ln4.

ln(x²)=2lnx only when x>0; for x≠0 the safe statement is ln(x²)=2ln|x|.