Questions tagged [logarithms]
Questions related to real and complex logarithms.
10,256
questions
4
votes
2
answers
287
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:
$$\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right)...
- 1,011
-2
votes
1
answer
62
views
What is the product of the solutions to the equation (√10)(x^(log (x))) = x^2? [closed]
I have spent a few hours on this question but I can't seem to grasp it. I have found that taking the log of both sides doesn't give me any progress. The log exponent identity didn't work. I also tried ...
4
votes
1
answer
178
views
Prove that sequence $(a_n)_{n\geq 1}$ is not convergent.
The standard branch of logarithm, $\log:\mathbb{C}\setminus (-\infty,0]\to\mathbb{C}$ is defined as
$$
\log(z):=\ln|z|+i\operatorname{Arg}(z)\tag{2}
$$
where $\arg(z)$ is the standard branch of the ...
- 936
0
votes
1
answer
41
views
Integral of Complex logarithm makes sense?
For example, I know that the principal branch of logarithm is not defined over negative real axis.
I think the integral of this logarithm along a circle doesn’t make sense.
Moreover, I know that the ...
- 11
1
vote
0
answers
67
views
What does "any polynomial dominates any logarithm" mean here?
My textbook states that
any polynomial dominates any logarithm: $n$ dominates $(\log n)^3$. This also means, for example, that $n^2$ dominates $n\log n$
However, it wasn't clear to me what the ...
- 3,019
-4
votes
0
answers
65
views
Logarithms find the solution
What conditions must $a$ and $b$ satisfy for the equation to have at least one real solution? Find all the solutions of this equation:
$1+\log_b(2\log(a)-x)\log_x(b)=2\log_b(x)$
I have tried ...
-1
votes
1
answer
100
views
Solve the power equation $\;2\!\cdot\!3^{x^2}=6^x$
Solve the power equation $\;2\!\cdot\!3^{x^2}=6^x.$
$$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
$$\beta_i = x_i \log \frac{...
- 400
2
votes
1
answer
54
views
Prove $\ln (1+x) \leq x - x^2/4 $ for $x \leq 1$ using Taylor's theorem
RTP: $\ln (1+x) \leq x - \frac{1}{4} x^2$ for $x \leq 1$
I am trying to prove this specifically using Taylor theorem. Here is what I have so far:
$\ln (1+x) = x - \frac{x^2}{2} + \frac{x^3}{3(1+\xi)^3}...
- 487
3
votes
3
answers
385
views
$\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$ as a limit of a sum
Working on the same lines as
This/This and
This
I got the following expression for the Dilogarithm $\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$:
$$\operatorname{Li}_{2} \left(\frac{1}{e^{\...
- 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$$
Why is modulus sign needed. If this is because the domain of logarithmic function is $(0,\infty)$ Then why don't we mention the limitations of the domains of other ...
- 199
-2
votes
0
answers
141
views
Solving $\sqrt{x+1}=-2$ and looking for complex solutions [duplicate]
So the question was
$$\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})}$$
Then ...
- 19
1
vote
1
answer
19
views
Rounding to nearest integer in the log domain
This is about the threshold of whether to round down to $N$ or round up to $N+1$ where the proportional rounding error (or the error in the log domain) is smallest.
What is the simplest way to prove
$$...
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