- #1
josephgerth
- 8
- 0
Homework Statement
Prove that $$log_{b}(xy)=log_{b}x+log_{b}y.$$
Homework Equations
Let $$b^{u}=x,b^{v}=y.$$ Then $$log_{b}x=u,log_{b}y=v.$$
The Attempt at a Solution
I'm afraid I've been using circular reasoning to prove this. I can get this to a point where I have $$log_{b}(b^{u+v})=log_{b}b^{u}+log_{b}b^{v},$$ but I don't have a good way to simplify either side. There is a property which I later prove for which $$log_{b}(b)^{a}=a.$$ But I need the proof of the product rule in order to prove this! Every proof I've referred to uses the exponent rule (just shown) to prove the product rule, but... That doesn't seem quite right. What am I missing?
Thanks for your time and help.