 The change of base property can be summarized in one equation: log a (x) = log b (x)/log b (a)  This property is often essential for simplifying logarithms.  In particular, we often have to use it to put logarithms in a form we can solve with our calculators.

 Let x = b y and let a = b z.  Raise each side of the second equation to the power (y/z).  By the properties of exponents, this gives us a y/z = b z(y/z), which is equivalent to a y/z = b y. We know that b y = x, so we substitute x into the equation.  We now have x = a y/z. By taking the log base a of both sides, we are left with log a (x) = y/z.  From the first two equations, y = log b (x) and z = log b (a). Thus, we can substitute for y and z, leaving log a (x) = log b (x)/log b (a).

 Using a calculator, calculate the value of log 5 9.

 Our calculator probably doesn’t have a function for logarithms of base 5, so we need to swap our base to something we can use, like e.  Using the change of base property, log 5 (9) = ln(9)/ln(5) ≈ 2.2/1.6 = 1.38.