Derivation logarithm change of base formula, common log to natural log, log change of base examples
We start with a quick derivation of the logarithm change of base formula. Like all the properties of logarithms, we can get a handle on the derivation by turning the equation around, going from logarithmic to exponential form. We give the right hand side a name: y=log_b(a), we invert the log equation, then we apply log base c to both sides, apply the exponent property of logarithms and quickly arrive at the change of base formula.
But why should we care about the change of base formula? One good reason is that many technologies don't understand logs other than logs with a base of e (natural logs). Using the change of base formula, we can reformulate any logarithmic expression in terms of natural logs. We start the log change of base examples with a common log example (log base 10), converting common log to natural log using the change of base formula.
In the next example, we convert log_2(5) to log base 5. This results in an interesting relationship: log_2(5)=1/log_5(2). After noticing the reciprocal relationship produced by swapping the base and the argument of the logarithm, we show that this works in general when we switch the base and the argument of a log!