# Is there a proof why this law for logarithm is such?

• Nov 16th 2010, 07:36 PM
dd86
Is 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 PM
Unknown008
What about putting log to base 10 and log to base e in front of both sides?
• Nov 16th 2010, 09:21 PM
dd86
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 PM
Unknown008
You can do it like this:

Let:

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

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

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

$\log\ 3 = \log\ x$

Which implies that x = 3
• Nov 16th 2010, 10:09 PM
dd86
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 PM
Unknown008
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 PM
Quote:

Originally Posted by dd86
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!

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 AM
HallsofIvy
One can either first define $10^x$ and $e^x$ and the define $log_{10}(x)$ and $ln(x)$ as their inverses or vice-versa. Either way $10^{log(x)}= x$, $log(10^x)= x$, $e^{ln(x)}= x$, and $ln(e^x)= x$ 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.