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Algebra II

Prove $\sum_{{n = 2}}^{\infty} \frac{1}{n \ln (n) \ln \ln n}$ is divergent.

Natasha Gill 2022-01-21

How do you simplify a log with an exponent in the base?
Suppose you had:
$\log_{x^{b}} (y)$
How can you simplify this? Do you use the change of base formula?
Note: I tried to come up with something similar to a homework problem without actually being a homework problem. I think this is the most simple form.

gea3stwg 2022-01-21

Joint Entropy closed-form analytical solution
Differential entropy of a single Gaussian random variable is
$H (X) = \frac{1}{2} \ln (2 π e σ 2)$
What then is the closed-form analytical solution for joint entropy, H(X,Y)?

David Rojas 2022-01-21

Log integrals I
In this example the value of the integral
$I_{3} = \int_{0}^{1} \frac{\ln^{3} (1 + x)}{x} d x$
was derived. The purpose of this question is to determine the value of the more general form of the integral, namely,
$I_{n} = \int_{0}^{1} \frac{\ln^{n} (1 + x)}{x} d x .$
Also is there a corresponding value for the integral
$J_{n} = \int_{0}^{1} \frac{\ln n (1 + x)}{1 + x} d x ?$

siutulysr5 2022-01-21

Prove by Mean Value Theorem $\frac{x}{1 + x} < \ln (1 + x) < x for x > 0$

PEEWSRIGWETRYqx 2022-01-21

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

widdonod1t 2022-01-21

Find $lim_{x \to - \infty} \frac{\ln (1 + 3^{x})}{\ln (1 + 2^{x})}$

zgribestika 2022-01-21

Product of logarithms, prove this identity.

$2 \log (a) \log (b) = \log (a b)^{2} - \log (a)^{2} - \log (b) 2$

for a>1and b>1?

David Young 2022-01-21

Show ${(\log n)}^{(\log n)} = 2^{(\log n) (\log (\log n))}$
I am having difficulty understanding how this follows.
${(\log n)}^{(\log n)} = 2^{(\log n) (\log (\log n))} = n^{\log \log n}$
Which logarithmic identities are used to go through each equality?
e.g. how do you first go from
${(\log n)}^{(\log n)} = 2^{(\log n) (\log (\log n))}$
and then to
$2^{(\log n) (\log (\log n))} = n^{\log \log n}$
(The log base must be 2 or else this equality won't hold)

Gwendolyn Willett 2022-01-21

Why are the domains for ${\ln x}^{2}$ and $2 \ln x$ different?

lugreget9 2022-01-21

A reason for the value of $\int_{0}^{1} \log (x) \log (1 - x) d x$

Lennie Davis 2022-01-21

Prove $\log_{\frac{1}{4}} \frac{8}{7} > \log_{\frac{1}{5}} \frac{5}{4}$

tebollahb 2022-01-21

Is there a different approach to evaluate $\int \ln (x) d x$ ?

burkinaval1b 2022-01-20

Why does logarithmic differentiation work even if logs are not defined for negative numbers?

William Boggs 2022-01-20

Prove that $\frac{x}{x + 1} < \ln (1 + x) < x$ for $x > - 1, x \neq 0$

Gregory Jones 2022-01-20

How to solve for negative numbers in logarithmic equations
I am trying to solve the equation
$z^{n} = 1 .$
Taking $\log$ on both sides I get $n \log (z) = \log (1) = 0$ .
$\Rightarrow n = 0$ or $z = 1$ .
But I clearly missed out $(- 1)^{even numbers}$ which is equal to $1$ .
How do I solve this equation algebraically?

Annette Sabin 2022-01-20

Evaluate the log expression
$\frac{1}{\log_{x y} (x y z)} + \frac{1}{\log_{y z} (x y z)} + \frac{1}{\log_{z x} (x y z)}$

pogonofor9z 2022-01-20

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

Adela Brown 2022-01-20

Iterative calculation of $\log x$
Suppose one is given an initial approximation of $\log x, y_{0},$ so that:
$y_{0} = \log x + ϵ \approx \log x$
Here, all that is known about x is that $x > 1$ . Is there a general method of improving that estimation using only addition & multiplication, i.e. without exponentiation or logarithms?
$y_{1} = f (y_{0}, x) = ?$

ajedrezlaproa6j 2022-01-20

$\log_{971} or \log_{861}$

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