Articles of logarithms

Showing if $n \ge 2c\log(c)$ then $n\ge c\log(n)$

Is this true that if $n \ge 2c\log(c)$ then $n\ge c\log(n)$, for any constant $c>0$? Here $n$ is a positive integer.

Show root test is stronger than ratio test

This question already has an answer here: Inequality involving $\limsup$ and $\liminf$: $ \liminf(a_{n+1}/a_n) \le \liminf((a_n)^{(1/n)}) \le \limsup((a_n)^{(1/n)}) \le \limsup(a_{n+1}/a_n)$ 1 answer

Question about fixpoints and zero's on the complex plane.

Define property $A$ for an entire function $f(z)$ as $1)$ $f(z)=0$ has exactly one solution being $z=0$ $2)$ $f(z)=z$ has exactly one solution $=>z=0$ (follows from $1)$ ) $3)$ $f(z)$ is not a polynomial. Define property $B$ for an entire function $f(z)$ as $1)$ $f(z)=f_1(z)$ with property $A$. $2)$ $f_i(z)= ln(f_{i-1}(z)/z)$ for every positive integer […]

How can I solve $8n^2 = 64n\,\log_2(n)$

I currently try to analyze the runtime behaviour of several algorithms. However, I want to know for which integral values $n$ the first algorithm is better ($f(n)$ is smaller) and for which the second one is better. The two algorithms: $$f(n) = 8n^2$$ $$g(n) = 64n\,\log_2(n)$$ I started by equaling the two to $8n^2 = […]

Limit $\lim_{x\to\infty} x-x^{2}\ln\left(1+\frac{1}{x}\right)$

This question already has an answer here: How do I determine $\lim_{x\to\infty} \left[x – x^{2} \log\left(1 + 1/x\right)\right]$? 4 answers

Is showing $\lim_{z \to \infty} (1+\frac{1}{z})^z$ exists the same as $\lim_{n \to \infty} (1+1/n)^n$ exists

My expanded question: Is showing $\lim_{z \to \infty} (1+\frac{1}{z})^z$ exists as $z$ goes through real values the same as $\lim_{n \to \infty} (1+\frac{1}{n})^n$ exists as $n$ goes through integer values? If not, how much additional work is needed to make the two equivalent? I am asking this because I had posted a question which stated […]

Proving a logarithmic inequality

I’m interested why this is true: $$ \text{Considering }\forall (x,y,z) \in (1,\infty) $$ The following holds: $$\log_xy^z+\log_x{z^y}+log_y{z^x} \geq \frac{3}{2}$$ This is taken from a high school textbook of mine. I tried finding a meaningful manipulation by using AM-GM, but that got pretty messy. I’d like to avoid Lagrange multipliers since this is meant to be […]

Showing $\log(2)$ and $\log(5)$

How do I show that: $$\log(2)=\sum^\infty_{n=1}(-1)^{n+1}\frac{1}{n}$$ and that $$\log(5)=\log(3)+\sum^\infty_{n=1}(-1)^{n+1}\frac{2^n}{n3^n}$$ Thanks in advanced.

Proof of concavity of log function

Does anybody have a proof of the concavity of the $\log{x}$ that does not use calculus?

How can I prove that $x-{x^2\over2}<\ln(1+x)$

How can I prove that $$\displaystyle x-\frac {x^2} 2 < \ln(1+x)$$ for any $x>0$ I think it’s somehow related to Taylor expansion of natural logarithm, when: $$\displaystyle \ln(1+x)=\color{red}{x-\frac {x^2}2}+\frac {x^3}3 -\cdots$$ Can you please show me how? Thanks.