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!

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- Nov 16th 2010, 07:36 PMdd86Is there a proof why this law for logarithm is such?
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! - Nov 16th 2010, 08:34 PMUnknown008
What about putting log to base 10 and log to base e in front of both sides?

- Nov 16th 2010, 09:21 PMdd86
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? - Nov 16th 2010, 09:26 PMUnknown008
You can do it like this:

Let:

$\displaystyle 10^{\log\ 3} = x$

$\displaystyle \log\ 10^{log\ 3} = \log\ x$

$\displaystyle \log\ 3 \times \log\ 10 = \log x$

$\displaystyle \log\ 3 = \log\ x$

Which implies that x = 3 - Nov 16th 2010, 10:09 PMdd86
Hmm that seems logical. Thanks for presenting a new perspective, then.

If any one else knows any other way(s) of solving this, I'd gladly welcome them to post more replies! - Nov 16th 2010, 10:16 PMUnknown008
Well, you can usually even say that the log and power 10 cancel, leaving 3, but that may be too direct for a question which asks you to evaluate this.

If for example you have a long number, were such a situation is involved, then you may use it. - Nov 16th 2010, 10:16 PMMacstersUndead
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. - Nov 17th 2010, 01:56 AMHallsofIvy
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.