I'd like to know whether there is a proof for the b/m laws. Not asking cos I've been asked to proved it.
10^log x = x
e^ln x = x
Thanks!
I did have a go at that, but it still doesn't make sense. I've looked through some web pages and all the books I have. They just state that it's a rule of logarithms.
One did give the proof in the following way:
Proof for x = 10^log x
Let x = 10^k (Eqn. 1)
Apply log to both sides:
log x = log 10 ^k
log x = k log 10
Since log 10 = 1, therefore k = log x
By substituting k= log x in Eqn. 1, we get x = 10^log x
The proof's quite clear, but what if I have been asked to evaluate 10^log 3? Do I just use the law without having to prove anything? Would I still get the same marks in a test compared to someone who can explicitly prove it?
This is analogous to showing that:
1) 10^x and log x are inverses, and
2) e^x and ln x are inverses.
((Recall that if f is an inverse of g, then f(g(x)) = g(f(x)) = x))
One way you can show that they are inverses is to show that their graphs are symmetric about the line x = y.
One can either first define $\displaystyle 10^x$ and $\displaystyle e^x$ and the define $\displaystyle log_{10}(x)$ and $\displaystyle ln(x)$ as their inverses or vice-versa. Either way $\displaystyle 10^{log(x)}= x$, $\displaystyle log(10^x)= x$, $\displaystyle e^{ln(x)}= x$, and $\displaystyle ln(e^x)= x$ follow from the definition of "inverse" function:
g(x) is the inverse function to f(x) if and only if both f(g(x))= x and g(f(x))= x.