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

1. ## 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!

2. What about putting log to base 10 and log to base e in front of both sides?

3. 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?

4. 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

5. 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!

6. 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.

7. 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.

8. 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.