# Is Logarithm Inequality True for 0 < a < b < 1?

• l-1j-cho
In summary, the log function is strictly increasing for any base greater than 1. However, if the base is in the range of 0 to 1, the function will be strictly decreasing. It is uncommon to use a logarithm with a base within this range.
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.

Hi l-1j-cho!

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

thank you!

take log b - log a and differentiate it.

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

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).

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

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

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:
$$\log_{\,1/G}\,(x) =\frac{\log\,_G\,(x)}{\log_{\,\,G}\,(1/G)}=-\log_{\,G}\,(x)$$

## 1. What is the basic concept of logarithm inequality?

The basic concept of logarithm inequality is that when two numbers are raised to the same power, the one with the larger base will result in a larger value. In other words, if a and b are both positive numbers less than 1, and a is smaller than b, then log(a) will be a larger negative number than log(b).

## 2. Why does logarithm inequality hold true for numbers between 0 and 1?

This is because the logarithm of a number between 0 and 1 will always be a negative number. As the base of the logarithm increases, the negative number will become more negative, resulting in a larger value overall.

## 3. Can you provide an example to demonstrate the logarithm inequality for 0 < a < b < 1?

For example, let a = 0.2 and b = 0.5. The logarithm of a is log(0.2) = -0.69897 and the logarithm of b is log(0.5) = -0.30103. Since a is smaller than b, we can see that log(a) is a larger negative number than log(b), proving the logarithm inequality in this case.

## 4. Does logarithm inequality hold true for numbers greater than 1?

No, logarithm inequality only holds true for numbers between 0 and 1. If the numbers are greater than 1, the opposite is true - the number with the larger base will result in a smaller value.

## 5. How is logarithm inequality used in scientific calculations?

Logarithm inequality is commonly used in scientific calculations to compare the relative size of numbers. It is especially useful when working with very large or very small numbers, as it allows for easier comparison and calculation. It is also used in fields such as biology and economics to measure growth rates and percent changes.

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