Laws of Logarithms There are very few laws of logarithms that let us work with them very effectively, despite the fact that logarithms are very hard to evaluate in general. Assume a> 0 is a (positive) real number. • log a (x)= y means x = a y for real numbers x> 0 and y. So: • log a (a x )= x for every real number • a log a (x) = x for every x> 0. The combination of the last two statements says that logs and exponential functions are inverse functions. However, you should be careful to keep track of when x must be strictly positive, since no logarithm can be defined at zero and the logarithm of a negative number is a complex number, which we also won’t deal with. The laws of exponents lead to the following laws of logarithms. Here we assume x and y are positive real numbers. 1. log a (xy) = log a (x) + log a (y) 2. log a  x y  = log a (x) − log a (y) 3. log a (x r )= r log a (x) for any real number r. We will generally only care about the case when a = e, in which case log a (x) = ln(x) is the natural logarithm. One excuse for only working with this most important logarithm is the following “change of base formula” (for a = 1) log a (x)= ln(x) ln(a) . For the natural log, the laws become: 1. ln(xy) = ln(x) + ln(y) 2. ln  x y  = ln(x) − ln(y) 3. ln(x r )= r ln(x) for any real number r. Also, the earlier statements become: • ln(x)= y means x = e y for real numbers x> 0 and y. • ln(e x )= x for every real number • e ln(x) = x for every x> 0. 1