Simple logarithem

1. Jul 1, 2011

l-1j-cho

if 0 < a < b < 1, is log(a) < log (b) true for all a and b?
I haven't found any counter example but just to make it sure.

2. Jul 1, 2011

I like Serena

Hi l-1j-cho!

Yes, it is actually true for any 0 < a < b, since the log function is a strictly increasing function.

3. Jul 2, 2011

thank you!

4. Jul 2, 2011

pessimist

take log b - log a and differentiate it.

5. Jul 2, 2011

Nanas

It is true for a<b and we know that the domain is (0,infinity) where the base is greater than 1 Because The exponetial function are increasing function with base greater than one and we know that logarithms are inverse functions of exponetial functions try to prove it.

Thanks

6. Jul 2, 2011

SammyS

Staff Emeritus
It depends upon what you mean by the log function -- what base is being used.

If 1 < G,
then log G (x) is strictly increasing on its entire domain, (0, +∞).

However, 0 < 1/G < 1 and we have log 1/G (x) is strictly decreasing on its entire domain, (0, +∞). You conjuncture would be false in this case​

So if by the log function, you mean loge a.k.a. ln, or if you mean log10, then the log function is strictly increasing.

If on the other hand, you are referring to the logarithm function generically, then your conjecture is true if and only if the base is greater than 1.

It is somewhat unusual to use a logarithm with a base in the range (0, 1).

7. Jul 2, 2011

Nanas

I didn't understand what you mean here.Why it is unusual ?

8. Jul 2, 2011

SammyS

Staff Emeritus
It's just unusual to see something like log 0.2 (6), for example.

If 1 < G, then 0 < 1/G < 1 .

Doing a change of base gives:
$$\log_{\,1/G}\,(x) =\frac{\log\,_G\,(x)}{\log_{\,\,G}\,(1/G)}=-\log_{\,G}\,(x)$$