Questions related to real and complex logarithms.

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3answers
112 views

How to prove that $\ln(1+x^2)<x$ , given that $x>0$

Given that $x>0$, how to prove that $\ln(1+x^2)<x$? I have been thinking about Taylor series, but didn't know how to do it. any suggestions?
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4answers
1k views

Using Taylor's Theorem to show that $\ln(1 + x^2) \leq x^2$

Can we show that if $\operatorname{abs}(x) \lt 1$, then $$\ln(1+x^2) \leq x^2\;,$$ using Taylor's Theorem? I am thinking of expanding it about $x=0$ but I got something like $$f(x) = -x^2 + ...
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2answers
75 views

Does the matrix logarithm always converge for exponential matrices?

We have defined functions on square matrices $X$: $$e^X := I + X + \dfrac{X^2}{2!}+\dfrac{X^3}{3!}$$ and $$log(X):= (X-I)-\dfrac{(X-I)^2}{2} + \dfrac{(X-I)^3}{3}-...$$ The exponential converges for ...
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1answer
38 views

Approximate log of the sum

Suppose I want to approximate the following sum: $\log( \sum_{n=1}^\infty s_n e^{X_{n}})$, where $(X_n)$ is linear. Is there any smart way to approximate the first sum non-numerically?
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0answers
38 views

Prob. 7, Chap. 1 in Baby Rudin

Here's problem 7 in the exercises following Chap. 1 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Fix $b > 1$, $y > 0$, and prove that there is a unique real number ...
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1answer
349 views

Understand Logarithm of Bar values manipulation step.

Currently I am learning Logarithm , but I can't understand the manipulation of the following Highlighted step how it comes How the result come after after ...
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2answers
37 views

Find the value of $\log_{xz}(xy^4z) \times \log_{xy}(xyz^4)$

if $x,y,z \gt 1$ and $x^2=yz$ find the value of $$E=\log_{xz}(xy^4z) \times \log_{xy}(xyz^4)$$ what i did is $$E=\log_{xz}(xy^4z) \times \log_{xy}(xyz^4)=(1+4\log_{xz}y)\times (1+4\log_{xy}z)$$ ...
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2answers
57 views

Solve for $x$ in $\frac{x}{\ln(x)}=a$. Why does Wolfram alpha use complex numbers here?

Is there any possible way of doing this without using complex numbers? And why are complex numbers used?
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2answers
34 views

Change of base proof without reciprocal

I am looking for a proof of the change of base formula without using the reciprocal. I know that: $$log_ax=\frac{log_bx}{log_ba}$$ The proof usually involves taking the reciprocal: $log_ax=y$ ...
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3answers
112 views

Prove the derivative of the natural logarithm using the limit definition.

I know how to prove that the derivative of $\ln(x)$ is ${1\over x}$ using the definition $f'(x) = {f(x+h) - f(x) \over h}$ but I have ran into trouble proving that the derivative of $\ln(f(x))$ is ...
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3answers
404 views

Series for logarithms

This is more of a challenge than a question, but I thought I'd share anyway. Prove the following identities, and prove that the pattern continues. \begin{equation*} ...
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5answers
747 views

Do these series converge to logarithms?

It is well known that $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}... =\log(2).$$ If we consider the array: $T(n,k) = -(n-1)\; \text{ if }\; n|k, \;\text{ else } \;1,$ Starting: ...
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10answers
1k views

How can I find the value of $\ln( |x|)$ without using the calculator?

I want to know if there is a way to find for example $\ln(2)$, without using the calculator ? Thanks
1
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0answers
39 views

A binary BBP-type formula for log(23)

This question is related to Is there a binary spigot algorithm for log(23) or log(89)? by Dan Brumleve. A binary BBP-type formula is a convergent series formula of the type $$ ...
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0answers
184 views

Why is $e$ close to $H_8$, closer to $H_8\left(1+\frac{1}{80^2}\right)$ and even closer to $\gamma+log\left(\frac{17}{2}\right) +\frac{1}{10^3}$?

The eighth harmonic number happens to be close to $e$. $$e\approx2.71(8)$$ $$H_8=\sum_{k=1}^8 \frac{1}{k}=\frac{761}{280}\approx2.71(7)$$ This leads to the almost-integer ...
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2answers
51 views

Are there other functions of sets $f$ such that they have this property?

$f(A \cap B) = f(A) \cup f(B)$ This function is similar to the $\log_c$ function in that application of it onto a multiplication is equivalent to the summation of its applications. $\log_c(ab) = ...
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0answers
57 views

Logarithms applied to aleph 0 [duplicate]

I know that Ln(aleph 1) = aleph0 So Ln(aleph 0) = ? What is that number? Thanks for attention, really. Cheers R. Aragón M.
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3answers
22 views

How to express $\log_3(2^x)$ using $\log_{10}$? And how to evaluate $4^{\log_4y}$?

How to express $\log_3(2^x)$ using $\log_{10}$? And how to evaluate $4^{\log_4y}$?
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3answers
89 views

Showing that the series $\sum \log{n}/n^2$ converges.

I aim to show that the series $$\sum \frac{\log{n}}{n^2}$$ converges. I know that the inequality $log(n) < \sqrt{n}$ holds for large $n$. So this give us one way to prove the convergence of $\sum ...
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1answer
42 views

Integrate $\int(\log(\sin x \cos x))^n dx$ with hypergeometric function form

Evaluate $$\int({\log(\sin x\cos x)})^{n} \, \mathrm{d}x$$ with result in hypergeometric function form Could anyone help me with that?
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0answers
17 views

A group with unusual discrete log properties.

Does there exist a group where computing $g^x$ from $g^{a^{x}}$ is easy, computing $g^{a^{x}}$ from $x$ and $g^{a}$ is hard, and computing $x$ from $g^a$ and $g^{a^x}$ is hard. Intuitively I would ...
7
votes
4answers
1k views

Limit to infinity with natural logarithms $\lim_{x\to \infty } \left(\frac{\ln (2 x)}{\ln (x)}\right)^{\ln (x)} $

I found the following problem in my calculus book: Solve: $$\lim_{x\to \infty } \left(\frac{\ln (2 x)}{\ln (x)}\right)^{\ln (x)} $$ I tried to solve it using log rules and l'Hôpital's rule with no ...
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1answer
48 views

Solving logarithmic equations without calculator

Hi I am stuck on this question $$ \log_x 10= 5 (\log_{10} x) +4 $$ The answer key gives the solutions $x = 10^{1/5}$ and $x = 1/10$.
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1answer
25 views

Logarithmic differentiation trouble with bottom of fraction

$y=\frac{(2x+3)^9}{\sqrt x(x^2-x)^6}$ I switched it to $\ln(y)=9\ln(2x+3)-6x^{1/2}\ln(x^2-x)$ and then used the log rules for derivatives I know and the product rule on the right side and wound up ...
3
votes
3answers
85 views

How to compute $\int_{0}^{(e-1)^2}{\ln(\sqrt{x}+1)} \,\mathrm dx $?

I have a problem with this integral. $$\int\limits_{0}^{(e-1)^2}\!\! \left({\ln(\sqrt{x}+1)} \right)\,\mathrm dx $$ I applied the substitution method $t = \sqrt{x}+1$, $2t = dx$ I changed integration ...
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3answers
62 views

Find the value of $3^{\log_4(5)} - 5^{\log_4(3)}$. [closed]

Find the value of $3^{\log_4(5)} - 5^{\log_4(3)}$. Is there any property that can help here?
7
votes
10answers
281 views

How to prove that $\log(x)<x$ when $x>1$?

It's very basic but I'm having trouble to find a way to prove this inequality $\log(x)<x$ when $x>1$ ($\log(x)$ is the natural logarithm) I can think about the two graphs but I can't find ...
2
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2answers
32 views

Positive entropy of system of mixed substance (mathematical viewpoint)

If two substances respectively having mass $m_1$ and $m_1$, constant pressure specific heat capacities $c_{p1}$ and $c_{p2}$, and temperatures $T_1$ and $T_1$ are mixed at constant pressure, reaching ...
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4answers
70 views

How do I find the critical points of this function involving e?

I have the function: $$g(x)={{1 \over \sqrt{2 \pi}} \cdot e^{{-(x-2)^2}/2}}$$ Through very tedious differntion, I got to: $$g'(x) = {{{-(x+2)} \cdot {e^{{-(x-2)^2}/2}}} \over {2 \pi}}$$ Setting ...
4
votes
1answer
432 views

An arctan series with a parameter

I'm trying to evaluate $$\sum_{n=1}^\infty \arctan \left(\frac{2a^2}{n^2}\right) \ , \ a >0. $$ The answer I get only seems to be correct for small values of $a$. What accounts for this? Using ...
0
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2answers
142 views

How to calculate 3x7 by using logarithm?

This is a story about Newton I read once when I was a child. Now that book is lost and I can only tell you what I remember. When Newton was young, he had been already famous in curiosity and ...
2
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1answer
44 views

What does Heron's Algorithm have to do with the construction of logarithmic tables

i need a little help answering this question, what does Heron's Algorithm have to do with the construction of logarithmic tables. I know that Heron's algorithm is used for finding square roots, but ...
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votes
2answers
1k views

Find intersection of linear and logarithmic lines

I have equations for two lines, one of which is linear and the other is logarithmic, ie: $$y = m_1 x + c_1$$ $$y = m_2 \cdot \ln(x) + c_2$$ ..and I need to find out where (if at all) these lines ...
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4answers
60 views

Paradox: Summation of natural logarithms

Consider the expression : $$\sum_{i=1}^{\infty}\ln(i+2)-\ln(i+4)$$ If one evaluates it out, one gets $$\ln(\frac{3\times4\times5\times6\times...}{5\times6\times7\times8\times...})=\ln(12)$$ That ...
16
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4answers
327 views

Yet another log-sin integral ${\large\int}_0^{\pi/3}\log(1+\sin x)\log(1-\sin x)\,dx$

There has been much interest to various log-trig integrals on this site (e.g. see [1][2][3][4][5][6][7][8][9]). Here is another one I'm trying to solve: $$\int_0^{\pi/3}\log(1+\sin x)\log(1-\sin ...
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1answer
30 views

Why does $\ln(1+\frac{3}{n^2}+o(\frac{1}{n^2}))=\frac{3}{n^2}+o(\frac{1}{n^2})$?

In order to show that a series converges, I want to show that $\sum\ln(\frac{v_{n+1}}{v_n})$ Which led me to the following first part of the equation, but I didn't achieved to solve it so I looked in ...
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3answers
617 views

How to calculate number of digits of a large number?

Does anyone know any efficient ways of finding the number of digits in the large number $N = 4^{4^{4^4}}$? Thanks.
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2answers
245 views

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

Prove the following limit without using approximations and derivatives: $$\lim_\limits{x\to -\infty}{\frac{\ln\left(1+3^{x}\right)}{\ln\left(1+2^{x}\right)}}=0$$ I cannot think of any possible ...
3
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2answers
66 views

I would like to calculate this limit: $ \lim_{n \to \infty}(n^2+1)\cdot(\ln(n^2-4)-2\ln(n)) $

I would like to calculate this limit: $$ \lim_{n \to \infty}(n^2+1)\cdot(\ln(n^2-4)-2\ln(n)) $$ but I am a bit lost on how to tackle the logarithm. Any help would be greatly appreciated.
3
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4answers
58 views

Is $\log(n!) \in\Theta(n \log n)$ [duplicate]

Is $\log(n!) \in\Theta(n \log n)$? I know it is $O(n \log n)$ because $\log(n!) \leq \log(n^n)$ which is the same as $\log(n!) \leq n \log n$. But how can I show it is also $\Omega(n \log n)$?
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5answers
46 views

logarithms properties

I know it's very easy and naive, but apparently I cannot understand the following equation. Can you please prove it? Thanks in advance. The equation is: $$2=3^{\frac{\ln{2}}{\ln{3}}}$$
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4answers
82 views

How do you solve the equation $0.5^x = 2^x + 3$?

I need help with the following problem: $0.5^x = 2^x + 3$ I know the answer is -1.72, but I have to explain step by step how to solve it and I'm not sure how. I know you're supposed to take the log ...
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1answer
24 views

Logarithm of complex matrix

For invertible matrix $A$, we have $\log(\det A) = \mathrm{tr}(\log A)$ due to a corollary of Jacobi's formula. What if we had the argument $iA$ instead? Would the above relation still hold? Edit: ...
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2answers
38 views

How to prove $cn < n^{\log_{2}n}$

How do you prove that for any given $c$, there exists an $n$ such that $$cn < n^{\log_{2}n}$$ ? I know that I have to write $n$ in terms of $c$, but I'm having trouble with the log in the exponent. ...
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1answer
23 views

Derivative of the inverse of exponential function a^x, with a>0 and a≠1

While studying exponential functions, I understood that $$\frac{d}{dx}a^x=(\ln a)a^x.$$ I also learned previously that if $g(x)$ is the inverse of $f(x)$, then the derivative of $g(x)$ and the ...
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1answer
40 views

Need to show the equality of logarithm.

To show: $ \lim_{n \rightarrow \infty} 2n ( a^{\frac{1}{2n}} -1)=log(a) $ for a>0. By definiton: $$ e^{x}=\lim_{m \rightarrow \infty}(1+ \frac{x}{xm} )^{xm} =: a $$ Now take the log of both sides: ...
0
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0answers
24 views

An…almost inverse Mellin Transform?

$$\log x= \frac{1}{2\pi i} \int_{\gamma-i\infty}^{\gamma+i\infty} \frac{\Gamma^2(-n)\Gamma(n+1)}{(x-1)^n\Gamma(-n+1)}\text{d}n$$ I thought that this was an Inverse Mellin Transform, but ultimately ...
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3answers
67 views

Which is asymptotically larger $n^2 \log(n)$ or $n (\log(n))^{10}$?

Which is asymptotically larger $n^2 \log(n)$ or $n ( \log(n))^{10}$? I have tried by plugging in the values and $n^2 \log(n)$ turns out to be bigger. How can this be done analytically?
0
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5answers
56 views

What is the Inverse function of $y = 10^{-x}$? Steps are appreciated.

What is the inverse of $y = 10^{-x}$? These are my steps for the problem. Step 1 $y = 10^{-x}$. Step 2 $x = 10^{-y}$ by inverse substitution. Step 3 $10^y(x) = 1$. Step 4 $10^y = ...
8
votes
7answers
2k views

An alternative way to calculate $\log(x)$

How can I replace the $\log(x)$ function by simple math operators like $+,-,\div$, and $\times$? I am writing a computer code and I must use $\log(x)$ in it. However, the technology I am using does ...