Unsolved Challenge: Natural logarithm and Exponent

In summary, the natural logarithm (ln(x)) is the inverse of the exponential function and is based on the mathematical constant e. It has an inverse relationship with the exponent function, "undoing" its effects. The natural logarithm differs from other logarithmic functions in its base and is used in various mathematical and scientific calculations. Its properties include ln(1) = 0, ln(e) = 1, and ln(xy) = ln(x) + ln(y), and it is also used in calculus through the logarithmic derivative. Both the natural logarithm and exponent have practical applications in fields such as finance, physics, biology, and economics, including compound interest, population growth, radioactive decay, and modeling exponential growth and decay
  • #1
anemone
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Prove $e^{-x}\le \ln(e^x-x-\ln x)$ for $x>0$.
 
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Partial answer

Let $ x = x_0\approx 0.5671432$ be the point at which $e^{-x} = x$ (in terms of the Lambert function, $x_0 = W_0(1)$). Then $\ln x_0 = -x_0$, and so $\ln(e^{x_0} - x_0 - \ln x_0) = \ln(e^{x_0}) = x_0 = e^{-x_0}$. Thus at the point $x_0$ the two functions $e^{-x}$ and $\ln(e^x - x - \ln x)$ are equal.

Now let $f(x) = \ln(e^x - x - \ln x)$. Then $$f'(x) = \frac{e^x - 1 - \frac1x}{e^x - x - \ln x} = \frac{-1 + \bigl(e^x - \frac1x\bigr)}{e^x - (x+\ln x)}.$$ When $x = x_0$, both of the expressions in parentheses in that last fraction vanish. Therefore $f'(x_0) = \dfrac{-1}{e^{x_0}} = -e^{-x_0}$, which is the same as the derivative of $e^{-x}$ at $x_0$.

When $x>x_0$, $e^x - \frac1x$ and $x + \ln x$ are both positive. So in the expression for $f'(x)$ the numerator is greater than $-1$ and the denominator is less than $e^x$, and so $f'(x) > -e^{-x}$. Conversely, when $x<x_0$, $e^x - \frac1x$ and $x + \ln x$ are both negative and so $f'(x) < -e^{-x}$. It follows that $x_0$ is a local minimum for the function $f(x) - e^{-x}$. Therefore $\ln(e^x - x - \ln x) \geqslant e^{-x}$ in the neighbourhood of $x_0$.

Once you get away from the neighbourhood of $x_0$ it ought to be relatively easy to see that $\ln(e^x - x - \ln x)$ is greater than $e^{-x}$, but I don't have the patience or energy to pursue that part of the problem.
 
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FAQ: Unsolved Challenge: Natural logarithm and Exponent

1. What is the natural logarithm and exponent?

The natural logarithm is a mathematical function that describes the growth rate of a phenomenon. It is the inverse of the exponential function, which represents the rapid growth of a quantity over time. The natural logarithm is denoted by "ln" and is the logarithm with base e, a mathematical constant approximately equal to 2.71828. The exponent is the number of times a base number is multiplied by itself. For example, in the expression 2^3, 2 is the base and 3 is the exponent.

2. What is the relationship between the natural logarithm and exponent?

The natural logarithm and exponent are inverse functions of each other. This means that the natural logarithm of a number is the exponent that the base e must be raised to in order to get that number. For example, ln(e) = 1, because e^1 = e. Similarly, e^ln(x) = x for any positive number x.

3. How are the natural logarithm and exponent used in real-life applications?

The natural logarithm and exponent have many practical applications in fields such as finance, biology, and physics. In finance, the natural logarithm is used to calculate compound interest and growth rates. In biology, it is used to model population growth and decay. In physics, it is used to describe exponential decay and growth in radioactive materials.

4. What are some properties of the natural logarithm and exponent?

The natural logarithm and exponent have several important properties. The natural logarithm of 1 is 0, and the natural logarithm of e is 1. The natural logarithm is an increasing function, meaning that as the input increases, the output also increases. The exponent function is a continuous function, meaning that it has no abrupt changes in value. Additionally, the natural logarithm and exponent have inverse properties, as mentioned in question 2.

5. Are there any unsolved challenges related to the natural logarithm and exponent?

Yes, there are still some open problems and unsolved challenges related to the natural logarithm and exponent. One such challenge is the Riemann Hypothesis, which states that all non-trivial zeros of the Riemann zeta function lie on the critical line with real part 1/2. This has implications for the distribution of prime numbers and has yet to be proven. Additionally, there are ongoing efforts to find closed-form solutions for certain exponential and logarithmic equations, which would have significant practical applications.

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