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l-1j-cho

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I haven't found any counter example but just to make it sure.

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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.

- #1

l-1j-cho

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I haven't found any counter example but just to make it sure.

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I like Serena

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MHB

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

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l-1j-cho

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thank you!

- #4

pessimist

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take log b - log a and differentiate it.

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Nanas

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Thanks

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SammyS

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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

However, 0 < 1/G < 1 and we have log

So if by the log function, you mean log

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).

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Nanas

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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 ?

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SammyS

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

Doing a change of base gives:

[tex]\log_{\,1/G}\,(x) =\frac{\log\,_G\,(x)}{\log_{\,\,G}\,(1/G)}=-\log_{\,G}\,(x)[/tex]

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).

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.

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.

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.

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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