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Questions tagged [logarithms]

Questions related to real and complex logarithms.

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What is $ \int_{0}^{\exp(-1)} \frac{\ln \ln \frac{1}{x}}{1+x^{2}} dx $?

Background According to p. 22 of the following paper by Blagouchine, we have the following Malmsten integral evaluation: $$ \int_{0}^{1} \frac{\ln \ln \frac{1}{x}}{1+x^{2}} dx = \frac{\pi}{2} \ln\left(...
Max Muller's user avatar
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-2 votes
0 answers
20 views

Harmonic Conjugate of $\log|z|$ on Annulus? [closed]

Does $\log|z|$ possesses any harmonic conjugate in the annulus $B(0,r,1)=\{z:r<|z|<1\}$? Or equivalently I want to know if there is any holomorphic branch of logarithm that exists on the ...
Ravi's user avatar
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4 votes
2 answers
306 views

Showing $\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right) \, dx=\frac{\pi^2 + 4}{2}$

While exploring possible applications for exponential substitution, I stumbled upon the following integral identity: $$\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right)...
Emmanuel José García's user avatar
-2 votes
1 answer
62 views

What is the product of the solutions to the equation (√10)(x^(log (x))) = x^2? [closed]

I have spent a few hours on this question but I can't seem to grasp it. I have found that taking the log of both sides doesn't give me any progress. The log exponent identity didn't work. I also tried ...
shivank chintalpati's user avatar
4 votes
1 answer
179 views

Prove that sequence $(a_n)_{n\geq 1}$ is not convergent.

The standard branch of logarithm, $\log:\mathbb{C}\setminus (-\infty,0]\to\mathbb{C}$ is defined as $$ \log(z):=\ln|z|+i\operatorname{Arg}(z)\tag{2} $$ where $\arg(z)$ is the standard branch of the ...
Max's user avatar
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0 votes
1 answer
41 views

Integral of Complex logarithm makes sense?

For example, I know that the principal branch of logarithm is not defined over negative real axis. I think the integral of this logarithm along a circle doesn’t make sense. Moreover, I know that the ...
Brody's user avatar
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1 vote
0 answers
67 views

What does "any polynomial dominates any logarithm" mean here?

My textbook states that any polynomial dominates any logarithm: $n$ dominates $(\log n)^3$. This also means, for example, that $n^2$ dominates $n\log n$ However, it wasn't clear to me what the ...
Princess Mia's user avatar
  • 3,019
-4 votes
0 answers
65 views

Logarithms find the solution

What conditions must $a$ and $b$ satisfy for the equation to have at least one real solution? Find all the solutions of this equation: $1+\log_b(2\log(a)-x)\log_x(b)=2\log_b(x)$ I have tried ...
alan centellas's user avatar
-1 votes
1 answer
100 views

Solve the power equation $\;2\!\cdot\!3^{x^2}=6^x$

Solve the power equation $\;2\!\cdot\!3^{x^2}=6^x.$ $$2 \cdot 3^{x^{2}} = 6^x $$ $$log_3(9) \cdot log_3(3^{x^{2}}) = log_3(6^x) $$ $$2x^2 - xlog_3(6) = 0 $$ $$x(2x - log_3(6)) = 0$$ $$x = 0$$ or $$...
Firefly's user avatar
  • 29
3 votes
1 answer
195 views

Generalizing a logarithmic inequality

Let $\{x_i\}_{i=1}^N$ and $\{y_i\}_{i=1}^N$ be real numbers in the interval (0,1). Define for each $i$ $$\alpha_i = x_i (1-x_i) \log^2 \frac{x_i (1-y_i)}{y_i (1-x_i)}$$ and $$\beta_i = x_i \log \frac{...
mikefallopian's user avatar
2 votes
1 answer
54 views

Prove $\ln (1+x) \leq x - x^2/4 $ for $x \leq 1$ using Taylor's theorem

RTP: $\ln (1+x) \leq x - \frac{1}{4} x^2$ for $x \leq 1$ I am trying to prove this specifically using Taylor theorem. Here is what I have so far: $\ln (1+x) = x - \frac{x^2}{2} + \frac{x^3}{3(1+\xi)^3}...
punypaw's user avatar
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3 votes
3 answers
385 views

$\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$ as a limit of a sum

Working on the same lines as This/This and This I got the following expression for the Dilogarithm $\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$: $$\operatorname{Li}_{2} \left(\frac{1}{e^{\...
Srini's user avatar
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3 votes
2 answers
279 views

A problem that could use substitution or logs, not sure which works better

This is one of those brain teaser problems on instagram, and it starts here: $$x^{x^2-2x+1} = 2x + 1$$ And we want to solve for x. My first instinct was to try this $$\ln(x^{x^2-2x+1}) = \ln(2x + 1)\\ ...
Jesse's user avatar
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3 votes
3 answers
104 views

Why to use modulus in integration of $1/x$ [closed]

$$ \int \frac1x = \log_e |x|+C$$ Why is modulus sign needed. If this is because the domain of logarithmic function is $(0,\infty)$ Then why don't we mention the limitations of the domains of other ...
ca_100's user avatar
  • 199
-2 votes
0 answers
141 views

Solving $\sqrt{x+1}=-2$ and looking for complex solutions [duplicate]

So the question was $$\sqrt{x+1}=-2$$ And obviously there is no value for it, However, If you do the thing with $e$ and $\ln{}$ $$e^{\ln{\sqrt{x+1}}}$$ and $$e^{\frac{1}{2}\cdot (\ln{x+1})}$$ Then ...
Jkt's user avatar
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1 vote
1 answer
19 views

Rounding to nearest integer in the log domain

This is about the threshold of whether to round down to $N$ or round up to $N+1$ where the proportional rounding error (or the error in the log domain) is smallest. What is the simplest way to prove $$...
robert bristow-johnson's user avatar
1 vote
1 answer
78 views

How to show $(\log_2 x)^4 \leq x^3$ for $x > 1$?

In Rosen's discrete Math textbook, they mention in the solutions for one problem that $(\log_2 x)^4 \leq x^3$ for $x > 1$. However, I'm not sure how to exactly derive that myself, nor does the ...
Bob Marley's user avatar
2 votes
0 answers
22 views

Find the point of intersection of an annuity and an investment

We were completing some classwork for Annuities and finding the Sum of a G.P for simple situations as well as finding the time take for an annuity to reach a given value. Given the context we wanted ...
Daniel Beadle's user avatar
0 votes
1 answer
41 views

How to solve for a value in a log

I have a formula: Weight=onerepmax*(0.488 + 0.538 * ln(-0.075*reps)) And I need to solve for reps given a onerepmax and a weight. I got as far as: ...
RobKohr's user avatar
  • 113
0 votes
1 answer
46 views

Determining the witnesses (constants $C_0$ and $k_0$) when showing $(log_b n)^c$ is $O(n^d)$ (b > 1 and c,d are positive)

I'm having a hard time finding the constants/witnesses $C_0$ and $k_0$ that show $(\log_b n)^c$ is $O(n^d)$. That is $|(\log_b n)^c| \leq C_0|n^d|$ for $n > k_0$ (b > 1, and c,d are positive). I ...
Bob Marley's user avatar
-5 votes
2 answers
85 views

If the domain of $f(x)$ is $(-3, 1)$, then what is the domain of $f(\ln x)$? [closed]

I need a clear explanation for this question: If the domain of $f(x)$ is $(-3, 1)$ then the domain of $f(\ln x)$ is ... a) $\;(e^{-1}, e^3)$ b) $\;(0, \infty)$ c) $\;(1, \infty)$ d) $\;(e^{-3}, e^...
Rit Mukherjee 's user avatar
-1 votes
3 answers
87 views

Is there any other function than $log$ for which $f(ab) = f(a)+f(b)$ and $f$ is monotonic? [closed]

Is there any other function than $log_{x}$ for which $f(ab) = f(a)+f(b)$ and $f$ is monotonic? If the answer to this question is yes, how can I find such functions?
mmh's user avatar
  • 243
0 votes
6 answers
195 views

How would you prove $\log_{2}x < \sqrt x$ for $x > 16$? [closed]

I'm not really showing how to prove this, since I tried finding the $x$-intercepts/zeros of $f(x) = \sqrt x - \log_{2} x$ , and see that $x = 4, 16$ work but inspection, but I'm not sure how to ensure ...
Bob Marley's user avatar
-3 votes
2 answers
191 views

How do you solve this equation $ \log_{2}(x) = \sqrt x$? [closed]

Disclaimer: Guys before voting to get the question closed I strongly feel we should instead have a feature on MSE that can merge such similar/duplicate questions since we got some really cool/through ...
Bob Marley's user avatar
2 votes
1 answer
90 views

Solving $\frac{\ln(y/x)}{y-x} = t$ for $x$ [duplicate]

I am having trouble solving an algebra formula which is for a project of mine. I must solve for $x$ ($y$ is a known value). $$\frac{\ln\left(\dfrac{y}{x}\right)}{y-x} = t$$ As I try to solve the ...
user1343039's user avatar
-4 votes
0 answers
108 views

Find $\arcsin c$, $\, c\in\Bbb C$ [duplicate]

A math fan sent me a solution of the weird equation $\sin z=2$ posted in Quora. It is Weird because in real calculus, we experienced that $-1\leq \sin x\leq 1$. I saw this question in so many places ...
Bob Dobbs's user avatar
  • 11.9k
-2 votes
1 answer
52 views

Why is logarithm with base 10 having number greater than 1 always positive? [closed]

Consider logarithm with base 10 and number k Why will it always be positive if k>1 and negative if k<1?
Anvi Mahajan's user avatar
1 vote
2 answers
77 views

How to calculate the limit $\lim_{x \to 0}\frac{\ln(1 + \sin(12x))}{\ln(1+\sin(6x))}$ without L'Hôpital's rule?

$$ \lim_{x \to 0}\frac{\ln(1+\sin12x)}{\ln(1+\sin6x)} $$ I know it's possible to calculate this limit just by transforming it; I think you need to use the knowledge that $$ \lim_{x \to 0}\frac{\ln(1+x)...
Maciej Miecznik's user avatar
2 votes
2 answers
62 views

Why the property of exponents holds true even for fractional powers

How can we prove that the property of exponents a^m × a^n =a^(m+n) (where "^" this sign denotes the power a is raised to)holds true even if m, n and a are fractions? Like I can clearly see ...
Shyam's user avatar
  • 49
1 vote
2 answers
59 views

Unexpected asymptotic logarithm behavior

I have recently seen a rather confusing asymptotic property of logarithms: $$ \log(n^4 + n^3 + n^2) \leq O(\log(n^3 + n^2 + n)) $$ I find this very unintuitive. Why would the log of a bigger ...
CharComplexity's user avatar
1 vote
0 answers
67 views

Why the log function is so important on the plane?

I am studying right now some Complex Analysis and I have seen the importance of the (complex) logarithm function in almost every subject in it. Now I'm intrigued with that (possible) relation between $...
underfilho's user avatar
0 votes
0 answers
46 views

When Does ((n^a)-1)/a)) Equal e; A Sophomore's plight

I am a high school student (sophomore) and have come across something I would like explained. I was watching 3blue1brown for an explanation of calculus, when he used the formula: lim a->0 (d/dx(n^x)...
Andrew Thorson's user avatar
1 vote
1 answer
41 views

What to consider when taking kth root on both sides of equality

Say I have the following expression: $10^{l} = a^{k}$ If I take the kth root of both sides, does that mean we get: $10^{\frac{l}{k}} = a$ We don't have to consider anything with plus or minus?
Bob Marley's user avatar
2 votes
1 answer
78 views

Solve the equation: $ \log_{\sin x} (\cos x) - 2 \log_{\cos x} (\sin x) + 1 = 0. $ [closed]

Solve the equation: $ \log_{\sin x} (\cos x) - 2 \log_{\cos x} (\sin x) + 1 = 0. $ Attempt: I transorm this equation in $(\log\cos x-\log\sin x)(\log\cos x+2\log\sin x)=0$, therefore $\cos x=\sin x$ ...
user avatar
3 votes
4 answers
165 views

Derivative of $e^{x+e^{x+e^{x+...}}}$

Let $y=$ $e^{x+e^{x+e^{x+...}}}$ To find $\frac{dy}{dx}$, I took the natural log on both sides, which gives: $$\ln y = x + e^{x+e^{x+e^{x+...}}}$$ Differentiating on both sides,$$\frac{1}{y}\frac{dy}{...
Haider's user avatar
  • 127
1 vote
1 answer
44 views

How to prove upper bound of this difference of the Sine Integral?

This exercise can be found in Mathematics LibreTexts (bottom of the page) . I have been stuck for about a day and have made minimal progress. Let $S(x)=\int_0^x\frac{\sin t}{t}$. Show that for $k \ge ...
Maxwell Nganyadi's user avatar
1 vote
1 answer
54 views

Logarithmic Function Calculation in Mathematica

I find these results in the evaluation of the logarithms that only differ in the sign $-$ I do not understand why in the first case $\operatorname{Log}[x+1]/8$ is not returned as an answer.
Emerson Villafuerte's user avatar
1 vote
1 answer
66 views

Construction of discontinuous $f$ such that $f(xy) = f(x)+f(y)$ [duplicate]

Question How to construct a discontinuous $f$ such that $f(xy) = f(x)+f(y)$. Domain of $f$ has to be some subset of $\mathbb{R}$ and range of $f$ is $\mathbb{R}$. Also, try to construct non ...
Debu's user avatar
  • 1
1 vote
2 answers
72 views

Log X to what base n yields a whole number [closed]

Does there always exist a real number 'n' such that $log_{n}x$ is a whole number for any real number x? If yes what would the function to find this number look like?
lylehunder's user avatar
0 votes
0 answers
34 views

Why is there no logarithmic form of the exponential distributive rule/power of a product rule?

When learning the laws of exponents and logarithms, one finds that there is a correspondence. Each law of exponents has a corresponding equivalent expression in terms of logarithms. For example, the ...
ziggurism's user avatar
  • 16.9k
4 votes
5 answers
113 views

Which one is closer to $3024^{2500}$? $10^{8000}$ or $10^{9000}$?

I first approached this question by applying log to $3024^{2500}$. $\log(3024^{2500}) = 8701.454467\cdots$ I then thought that since $8701$ is closer to $9000$, $3024^{2500}$ is closer to $10^{9000}$...
Nighty's user avatar
  • 2,198
0 votes
0 answers
40 views

When does $x\ln f(x)$ become convex?

When a function $f>0$ is defined on $x\geq 0$, I would like to know the conditions for $F(x)=x\ln f(x)$ to be convex. Naively, $f$ being convex looks sufficient, but it is not true even if $f$ is ...
Sakai's user avatar
  • 21
3 votes
1 answer
62 views

Euler Sums of Weight 6

For the past couple of days I have been looking at Euler Sums, and I happened upon this particular one: $$ \sum_{n=1}^{\infty}\left(-1\right)^{n}\, \frac{H_{n}}{n^{5}} $$ I think most people realize ...
Jessie Christian's user avatar
11 votes
2 answers
680 views

Implicit function equation $f(x) + \log(f(x)) = x$

Is there a function $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that $$ f(x) + \log(f(x)) = x $$ for all $x \in \mathbb{R}_{>0}$? I have tried rewriting it as a differential equation ...
Strichcoder's user avatar
  • 2,005
0 votes
1 answer
33 views

Natural Log's Property Doesn't Transfer Over

I am trying to rewrite the summation of $\ln(x)$ equation into a continuous function using logarithmic properties. We already know that $\left(\sum_{n=1}^{x}\ln\left(n\right)\right)$ is just equal to $...
Monke's user avatar
  • 1
2 votes
0 answers
100 views

Is ln|x| + C really the most general antiderivative of 1/x? [duplicate]

I recently stumbled across a claim that $\ln |x| + C$ isn't the most general antiderivative of $1/x$. The argument was that the parts of the curve $\ln |x|$ separated by the $y$-axis do not have to be ...
Alice's user avatar
  • 508
0 votes
2 answers
36 views

Logarithm approximation loses solution

I was doing an exercise in which I had to plot by hand a function. The function was $$f(x)=\ln\left(\frac{\sqrt[3]{x}}{3x - 1}\right)$$ which I rewrote into $$f(x)=\frac{\ln\left(x\right)}{3} - \ln\...
serax's user avatar
  • 135
0 votes
0 answers
21 views

Inequality for log-convex functions

Let $s:\mathbb{R}^+\to\mathbb{R}^+$ be a positive, decreasing, log-convex function such that $s(0)=1$, $\lim_{x\to\infty}s(x) = 0$. In addition, $-s'$ is also assumed to be log-convex. Are these ...
hugues_myr's user avatar
2 votes
0 answers
40 views

How is the dilogarithm defined?

I am pretty happy with the definition of the "maximal analytic contiuation of the logarithm" as $$ \operatorname{Log}_{\gamma}\left(z\right) = \int_\gamma ...
Jack's user avatar
  • 424
3 votes
2 answers
204 views

Why does the scalar inside a natural log dissapear when differentiating it? [closed]

For example if I was differentiating $\ln(2x)$ doesn't the chain rule dictate that it should be $2/x$, not $1/x$? Why does the $2$ disappear?
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