Equality of natural ln function

MHB
Thread starter tmt1
Start date Jul 16, 2016
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Function Ln Natural

In summary, the natural logarithm function, denoted as ln(x), is the inverse of the exponential function and calculates the power to which the base number e must be raised to obtain a given number. It differs from other logarithms by using the base e, which is useful in many mathematical and scientific applications. Its domain is all positive real numbers and its range is all real numbers. The natural logarithm function is related to exponential functions as its inverse, "undoing" the exponential function. It is commonly used in scientific research to model growth and decay, transform data, and has applications in calculus and differential equations.

Jul 16, 2016

tmt1

I have this equality:

$$ (\ln\left({n}\right))^4 < {n}^{\frac{1}{4}} $$ where $ n > 1$

Can I derive a law from this such that

$$ (\ln\left({n}\right))^b < {n}^{\frac{1}{b}} $$ where $n > 1$ ?

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Jul 16, 2016

Greg

Gold Member

MHB

Hi tmt. $\ln^4(n)<n^{1/4}$ is only true for $n\in\{1,2\}$, if $n\in\mathbb{N}$.

By the way, it's an inequality.

Last edited: Jul 16, 2016

1. What is the natural logarithm function?

The natural logarithm function, denoted as ln(x), is the inverse of the exponential function. It is a mathematical function that calculates the power to which the base number e (approximately equal to 2.71828) must be raised to obtain a given number.

2. How is the natural logarithm different from other logarithms?

The natural logarithm is different from other logarithms because it uses the base number e instead of other commonly used bases such as 10 or 2. This base is especially useful in many mathematical and scientific applications.

3. What is the domain and range of the natural logarithm function?

The domain of the natural logarithm function is all positive real numbers (x > 0). The range of the function is all real numbers (y ∈ ℝ).

4. How is the natural logarithm function related to exponential functions?

The natural logarithm function is the inverse of the exponential function. This means that if we have an equation in the form of y = e^x, the inverse function would be x = ln(y). In other words, the natural logarithm function "undoes" the exponential function.

5. How is the natural logarithm function used in scientific research?

The natural logarithm function is used in various scientific fields, such as biology, chemistry, and physics. It is commonly used to model exponential growth and decay, as well as to transform data that is not normally distributed into a more symmetrical distribution for statistical analysis. It also has applications in calculus and differential equations.