Questions tagged [logarithms]
Questions related to real and complex logarithms.
10,256
questions
4
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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:
$$\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right)...
- 1,011
-2
votes
1
answer
62
views
What is the product of the solutions to the equation (√10)(x^(log (x))) = x^2? [closed]
I have spent a few hours on this question but I can't seem to grasp it. I have found that taking the log of both sides doesn't give me any progress. The log exponent identity didn't work. I also tried ...
4
votes
1
answer
178
views
Prove that sequence $(a_n)_{n\geq 1}$ is not convergent.
The standard branch of logarithm, $\log:\mathbb{C}\setminus (-\infty,0]\to\mathbb{C}$ is defined as
$$
\log(z):=\ln|z|+i\operatorname{Arg}(z)\tag{2}
$$
where $\arg(z)$ is the standard branch of the ...
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votes
1
answer
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Integral of Complex logarithm makes sense?
For example, I know that the principal branch of logarithm is not defined over negative real axis.
I think the integral of this logarithm along a circle doesn’t make sense.
Moreover, I know that the ...
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1
vote
0
answers
67
views
What does "any polynomial dominates any logarithm" mean here?
My textbook states that
any polynomial dominates any logarithm: $n$ dominates $(\log n)^3$. This also means, for example, that $n^2$ dominates $n\log n$
However, it wasn't clear to me what the ...
- 3,019
-4
votes
0
answers
65
views
Logarithms find the solution
What conditions must $a$ and $b$ satisfy for the equation to have at least one real solution? Find all the solutions of this equation:
$1+\log_b(2\log(a)-x)\log_x(b)=2\log_b(x)$
I have tried ...
-1
votes
1
answer
100
views
Solve the power equation $\;2\!\cdot\!3^{x^2}=6^x$
Solve the power equation $\;2\!\cdot\!3^{x^2}=6^x.$
$$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$$ or $$...
- 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
$$\beta_i = x_i \log \frac{...
- 400
2
votes
1
answer
54
views
Prove $\ln (1+x) \leq x - x^2/4 $ for $x \leq 1$ using Taylor's theorem
RTP: $\ln (1+x) \leq x - \frac{1}{4} x^2$ for $x \leq 1$
I am trying to prove this specifically using Taylor theorem. Here is what I have so far:
$\ln (1+x) = x - \frac{x^2}{2} + \frac{x^3}{3(1+\xi)^3}...
- 487
3
votes
3
answers
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views
$\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$ as a limit of a sum
Working on the same lines as
This/This and
This
I got the following expression for the Dilogarithm $\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$:
$$\operatorname{Li}_{2} \left(\frac{1}{e^{\...
- 862
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
$$\ln(x^{x^2-2x+1}) = \ln(2x + 1)\\
...
- 2,692
3
votes
3
answers
103
views
Why to use modulus in integration of $1/x$ [closed]
$$ \int \frac1x = \log_e |x|+C$$
Why is modulus sign needed. If this is because the domain of logarithmic function is $(0,\infty)$ Then why don't we mention the limitations of the domains of other ...
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0
answers
141
views
Solving $\sqrt{x+1}=-2$ and looking for complex solutions [duplicate]
So the question was
$$\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})}$$
Then ...
- 19
1
vote
1
answer
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views
Rounding to nearest integer in the log domain
This is about the threshold of whether to round down to $N$ or round up to $N+1$ where the proportional rounding error (or the error in the log domain) is smallest.
What is the simplest way to prove
$$...
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 ...
- 883
2
votes
0
answers
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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 ...
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:
...
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1
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46
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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 ...
- 883
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votes
2
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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^...
-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?
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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 ...
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2
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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 ...
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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 ...
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-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 ...
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1
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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?
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vote
2
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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
$$
\lim_{x \to 0}\frac{\ln(1+x)...
2
votes
2
answers
62
views
Why the property of exponents holds true even for fractional powers
How can we prove that the property of exponents a^m × a^n =a^(m+n) (where "^" this sign denotes the power a is raised to)holds true even if m, n and a are fractions? Like I can clearly see ...
- 49
1
vote
2
answers
59
views
Unexpected asymptotic logarithm behavior
I have recently seen a rather confusing asymptotic property of logarithms:
$$
\log(n^4 + n^3 + n^2) \leq O(\log(n^3 + n^2 + n))
$$
I find this very unintuitive. Why would the log of a bigger ...
1
vote
0
answers
66
views
Why the log function is so important on the plane?
I am studying right now some Complex Analysis and I have seen the importance of the (complex) logarithm function in almost every subject in it. Now I'm intrigued with that (possible) relation between $...
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0
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When Does ((n^a)-1)/a)) Equal e; A Sophomore's plight
I am a high school student (sophomore) and have come across something I would like explained.
I was watching 3blue1brown for an explanation of calculus, when he used the formula: lim a->0 (d/dx(n^x)...
1
vote
1
answer
41
views
What to consider when taking kth root on both sides of equality
Say I have the following expression:
$10^{l} = a^{k}$
If I take the kth root of both sides, does that mean we get:
$10^{\frac{l}{k}} = a$
We don't have to consider anything with plus or minus?
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votes
1
answer
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Solve the equation: $ \log_{\sin x} (\cos x) - 2 \log_{\cos x} (\sin x) + 1 = 0. $ [closed]
Solve the equation:
$ \log_{\sin x} (\cos x) - 2 \log_{\cos x} (\sin x) + 1 = 0. $
Attempt: I transorm this equation in $(\log\cos x-\log\sin x)(\log\cos x+2\log\sin x)=0$, therefore $\cos x=\sin x$ ...
user1316790
3
votes
4
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Derivative of $e^{x+e^{x+e^{x+...}}}$
Let $y=$ $e^{x+e^{x+e^{x+...}}}$
To find $\frac{dy}{dx}$, I took the natural log on both sides, which gives: $$\ln y = x + e^{x+e^{x+e^{x+...}}}$$
Differentiating on both sides,$$\frac{1}{y}\frac{dy}{...
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1
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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 ...
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.
1
vote
1
answer
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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 ...
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2
answers
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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?
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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 ...
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5
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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 $10^{9000}$...
- 2,198
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0
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40
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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 ...
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votes
1
answer
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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 ...
- 203
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votes
2
answers
679
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 ...
- 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 $...
- 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 ...
- 508
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votes
2
answers
35
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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 $$f(x)=\frac{\ln\left(x\right)}{3} - \ln\...
- 135
0
votes
0
answers
21
views
Inequality for log-convex functions
Let $s:\mathbb{R}^+\to\mathbb{R}^+$ be a positive, decreasing, log-convex function such that $s(0)=1$, $\lim_{x\to\infty}s(x) = 0$. In addition, $-s'$ is also assumed to be log-convex.
Are these ...
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votes
0
answers
40
views
How is the dilogarithm defined?
I am pretty happy with the definition of the "maximal analytic contiuation of the logarithm" as
$$
\operatorname{Log}_{\gamma}\left(z\right) =
\int_\gamma ...
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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?
user1329181
8
votes
1
answer
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Approximating $\log x$ by a sum of power functions $a x^b$
Let's approximate $\log x$ on the interval $(0,1)$ by a power function $a x^b$ to minimize the integral of the squared difference
$$\delta_0(a,b)=\int_0^1\left(\log x-a x^b\right)^2dx.\tag1$$
It's ...
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vote
3
answers
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How does $\frac1{f(x)}\cdot\frac d{dx}f(x) = \frac d{dx}[\ln f(x)]$?
Perhaps a basic question, as I was told that this was easily solvable using the chain rule, although I can't figure out why:
$\frac1{f(x)}\cdot\frac d{dx}f(x) = \frac d{dx}[\ln f(x)]$
I am wondering ...
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