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Algebra II

Integral of $\int \frac{1}{\log x} d x$

Franco Franco 2022-01-22

Proving or disproving: If $0 < a < b < 1$ , then ${(1 - a)}^{b} > {(1 - b)}^{a}$

Aaden Lam 2022-01-22

Comparing Powers of Different Bases
How can I know if one power is bigger than the other when the bases are different?
For example, considering $2^{10}$ and $10^{3}$ the former is the greater one, but how to prove this? Logarithms? I'll be working with big numbers, and though a more general solution is really appreciated, I will be comparing exactly powers of $2$ and $10$ .

Abbie Mcgrath 2022-01-22

If x is rational, can $\frac{\log (1 - x)}{\log x}$ be algebraic?

iocasq4 2022-01-22

Use the unique factorization for integers theorem and the definition of logarithm to prove that $\log_{3} (7)$ is irrational.

Gwendolyn Giles 2022-01-22

Simple algebra clarification needed for Gaussian entropy formula
Norwich (2003) gives the following equality based on a substitution
$H = \frac{1}{2} \ln (2 π σ^{2}) + \frac{1}{2}$

Anika Osborne 2022-01-22

Break up the following function using laws of logarithms, then compute the derivative. $f (x) = \ln (\frac{6 (x + 1)}{x^{3}})$

Rudy Koch 2022-01-22

Express $\ln x y^{2}$ as a sum, difference or multiple of logarithms.

Yahir Haas 2022-01-22

Prove that
$- 2 \log (2) = - 2 + \sum_{n = 1}^{\infty} \frac{1}{n (2 n + 1)}$

Laney Spears 2022-01-22

How to prove monotonicity of this function?
Let $a > 0$ . How to prove that the function:
$f (x) = \frac{a x - 1}{\log (a x)} \cdot \frac{\log x}{x - 1},$
is monotonic (depending on $a < 1$ or $a > 1$ ). I know that we can calculate the derivative and determine its sign, but this needs much of calculation. I'm wondering if we can decide the monotonicity using a simple trick.

Margie Marx 2022-01-22

Derive $\log (a + b) = \log (a) - 2 \log (\cos (\arctan (\sqrt{\frac{b}{a}})))$

Jazmin Perry 2022-01-22

Show that $\int_{1}^{0} \frac{\ln (1 + x)}{x} d x = - \frac{1}{2} \int_{0}^{1} \frac{\ln x}{1 - x} d x$ without actually evaluating both integrals

Jazmin Perry 2022-01-22

Which is bigger among $(i) \log_{23}$ and $\log_{35} (i i) \log_{23}$ and $\log_{311}$ .

Laney Spears 2022-01-22

Find $n$ satisfying the equation $[\log_{21}] + [\log_{22}] + [\log_{23}] + \dots [\log_{2} n] = 1538$

taoigas 2022-01-22

A closed form for $\int_{0}^{1} \frac{\ln (- \ln x) l i^{2} x}{x} d x$

David Lewis 2022-01-22

Solve $- B \ln y - A y \ln y + A y - A = 0$ for $y$

fanyattehedzg 2022-01-22

Information on the sum $\sum_{n = 1}^{\infty} = \frac{\log n}{n!}$

PEEWSRIGWETRYqx 2022-01-22

How do I show this inequality that
$\frac{\ln (x + 1)}{\ln (x)} \leq \frac{x}{x - 1}$ for $x > 1$ ?

untchick04tm 2022-01-22

How to solve the following logarithmic equation: $n (n - 1) 3^{n} = 91854$

Ernest Ryland 2022-01-22

Infinite sum of logs puzzle
Here is a neat infinite sum puzzle:
Prove the following is true when $| x | < 1$
$- \ln (1 - x) = \ln (1 + x) + \ln (1 + x^{2}) + \ln (1 + x^{4}) \dots \ln (1 + x^{2 k}) \dots$
$- \ln (1 - x) = \sum_{k = 0}^{\infty} \ln (1 + x^{2 k})$

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Algebra II

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