Questions related to real and complex logarithms.

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0
votes
1answer
45 views

How do I prove that the only possible function is $exp$?

Let´s say we have a differentiable function $f : \mathbb{R} -> \mathbb{R}$ with $f' = f$ and $f(0) = 1$ . How do I show that the only possible function for this to work $f = exp$ ? ...
0
votes
1answer
30 views

If $m \ge 8 s \log(m^2 s)$, how much greater $m$ is relatively to $s$?

Given that $m,s \in \mathbb{N}$, if $m \ge 8 s \log(m^2 s)$, how much greater $m$ is relatively to $s$ ? It seems to me $m>>s$, but I would like some idea of the magniture. I'm not quite sure ...
2
votes
0answers
71 views

Solve for $x: \ln(x+4)+\ln(x-2)=5$

Solve for x: $\ln(x+4)+\ln(x-2)=5$ Where do I go from here? If there weren't four terms in the equation I would use the quadratic formula. How can I solve for x? EDIT 1: Is this correct? ...
-2
votes
1answer
33 views

How to define human-friendly axis marking notches

I have a need to compute values of "tick" marks on an axis that uses various different min/max ranges. Specific example -- Picking Human-Friendly Notch Step Axis Y has function values range between ...
3
votes
1answer
62 views

Solving $\log_2(x^2) = x$ explicitly?

I'm having problems getting a proper step-by-step solution to this equation. $$ \log_2(x^2) = x $$ I know the results are 2 and 4, but so far I can get only solutions like these: $$ 2^x = x^2 \qquad ...
-1
votes
1answer
40 views

Is this logarithmic inequality true?

Assume we have two complex variables $h_i$ and $h_d$ which satisfy the following relationship $$ 2\ |h_i|^2\leq \ |h_d|^2$$ can we say that $$\log\left( 1+ \frac{\big||h_d| - ...
4
votes
1answer
296 views

inequality $10<2^{2^{\frac {3}{\log_2 \log_2 10}}}$

While working on this question I ended up with $10<2^2{^{\frac {3}{\log_2 \log_2 10}}}$ I am looking for answers using methods similar to this or this or this or this. Alternative original ...
0
votes
0answers
35 views

Determine if the graph $f(x) = \ln(x)$ has any critical numbers

Determine if the graph $f(x) = \ln(x)$ has any critical numbers. The derivative would be $f'(x) = \frac{1}{x}$
1
vote
0answers
28 views

Logarithmic Series [duplicate]

I was doing a bit of math when I came across logarithmic series. I have no idea from where they come from. They seem so unrelated, that I have no intuition behind them at all. So, can anyone prove ...
0
votes
1answer
53 views

question about logarithms [closed]

Is it true that for $n\ge 2$ integer and $x\ge 1$ real, we have $\lfloor \log_nx \rfloor=\lfloor\log_n \lfloor x \rfloor \rfloor$ ?
1
vote
1answer
29 views

Find the point where the slope changes drastically

I have a distribution for which I have to find the point where the slope changes drastically. In visual terms, I have to find this point: I though I could use derivatives, but for the following ...
0
votes
1answer
54 views

Transformation Ricker equation

The classical Ricker equation for modelling density-dependent population growth is: $N_{t+1} = N_t * e^{r * \left(1-\frac{N_t}{k}\right)}$ where $N_t$ is the initial number of individuals (starting ...
2
votes
1answer
69 views

Why the answer to $\log(x+3)+\log(x+2)\geq \log 12$ different from that of $\log(x^2+5x+6)\geq \log12$?

When I have problem with $\log_b(x^2+2ax+a^2)\geq\log_bk$ I usually just say that $x^2+2ax+a^2\geq k$ and $x^2+2ax+a^2>0$. I used in example a $(x+a)^2$ to make the things easy. So I do the ...
0
votes
1answer
86 views

How to continue solving? Perfect Cuboid

I am doing research on perfect cuboids, and I'm looking for values $a,b,c$ such that the following is integer, and I'm not sure how to continue this. Any suggestions are appreciated! $PED$ is a very ...
4
votes
2answers
82 views

Why do you reject negative base solution for Logs?

$log_x64=2$ translates to $x^2=64$ This solves to $x=\pm8$ Why do you reject the solution of $x=-8$ ? Doesn't it successfully check? $log_{-8}64=2$ means "The exponent for -8 to get 64 is 2" which ...
0
votes
1answer
43 views

Solving logarithmic equation for $n$

I have the following equation and I am trying to isolate $n$: $$8n^2 = 64 n\log_2 n$$ Haven't done algebra in years and can't figure out how to get rid of the $\log_2$.
0
votes
1answer
25 views

How would you solve this logarithmic equation with both $\log n$ and $n$ terms?

I have a small logarithm related question that I do not seem to understand how to solve. If a program takes $\log n$ microseconds to run a program of size $n$, what is the maximum size of a program ...
1
vote
1answer
31 views

Can anyone help me in proving this?

As we know $e^{\ln (x)}=x$. To prove this I applied $\ln$ on LHS (left hand side) and I got RHS (rights hand side ) as follows $\ln (e^{\ln (x)})=\ln (x)$ $\ln (x)\ln (e)=\ln (x)$ $\ln (x)=\ln ...
1
vote
2answers
118 views

Intersection of functions $\ln(x)$ and $\frac{1}{x}$

How to find $x$ such that $$\ln(x)=\frac{1}{x}$$ Thank you!
1
vote
2answers
40 views

Investment question using logarithm

8,000 dollars is invested in an account that yields 6% interest per year. After how many years will the account be worth $14 000, to the nearest half year, if the interest is compounded monthly? ...
1
vote
2answers
71 views

Show that $(\ln a)^k \neq k \ln a $ [closed]

I have a question that I am not sure how to answer: Show that $(\ln a)^k \neq k \ln a $
0
votes
3answers
55 views

Solve logarithmic equation for $x$ to find the inverse of $f(x)= \ln(x+\sqrt{x^2+1})$

Let $f(x)= \ln(x+\sqrt{x^2+1})$. Find $f^{-1}(x)$. Here is what I got so far: $y= \ln(x+\sqrt{x^2+1})$, rewrite as $x= \ln(y+\sqrt{y^2+1})$, then $$e^x= y+\sqrt{y^2+1}$$ $$e^x-y= \sqrt{y^2+1}$$ ...
0
votes
1answer
40 views

How to compute antilogarithmic and superlogarithmic spaced values?

Let's suppose I have a range, e.g. $[100, 900]$. I want to compute 8 logarithmic spaced values $x_i={100, ..., 900}$. I use the following formula: $$x_1=\log(S)+\frac{(i-1)\log(S/L)}{n-1}$$ In the ...
0
votes
2answers
46 views

logarithm problems [closed]

How to carry out logarithm problems with a negative input?
6
votes
4answers
120 views

Are logarithms the only continuous function on $(0, \infty)$ that has this property?

Are logarithms the only continuous function on $(0, \infty)$ that has this property? $$ f(xy) = f(x) + f(y) $$ If so, how would we show that? If not, what else would we need to show that a function ...
1
vote
1answer
38 views

Analytical proof of the inequality $p^n (1-p\ln p)<1$ where $0<p<1$

Simulations with Mathematica suggest to me that $$p^n (1-p\ln p) <1\quad\text{for } p \in (0,1) \text{ and }n \ge 0$$ Have you got any hints on how I can derive an analytical proof of this ...
0
votes
0answers
32 views

On $p^{\log_q n}$, where $p$ and $q$ are distinct primes

Let $p,q$ be distinct primes, $n>1$ an integer with $\log_q n $ irrational. It was, and probably still is, a conjecture that $p^{\log_q n}$ is non-integer. What progress has been made towards it?
1
vote
4answers
51 views

help understanding how $\ln$ and $e$ cancel.

I realise cancel may be the wrong term and inverse may be more appropriate but these is one situation I really don't get…or rather haven't found a suitable explanation. Most sources I have come across ...
1
vote
0answers
21 views

Discrete logarithm - factorization of modul

I am solving discrete logarithm problem $mod N$. $N$ is composite number, i found its factors - lot of small primes and two big primes ($> 2^{50}$). Does the factorization of $N$ somehow help me? I ...
0
votes
1answer
24 views

Find roots for an equation with quadratic, linear and log terms?

I'm wondering if there exists a closed-form or analytic expression for the roots of an equation of the form $ax^2 + bx + c\log x=0.$ considering the natural $\log$. Wolfram alpha is leading me to ...
3
votes
2answers
91 views

Is it possible to solve this equation with logarithms and exponents?

$$-\frac{1}{3}\log(4x-12)+6=\left(-\frac{1}{2}\right)^x $$ Out of all the logarithm laws I've learned (which is pretty limited), I have not found a way to solve for what x is yet. Can someone verify ...
3
votes
1answer
60 views

Do these ratios of the Eulerian number triangle converge to the logarithm of x?

Consider the matrix $A_3$ with the definition if $n=k$ then $A_3(n,k)=\binom{n-1}{k-1}=1$, else if $n\ge k$ then $A_3(n,k)=\frac{\binom{n-1}{k-1}}{1-x}$ else $A_3(n,k)=0$. $\binom{n-1}{k-1}$ means the ...
0
votes
0answers
29 views

Properties of log map on matrices in $SE(3)$

I am learning about the log map on $SE(3)$ and I want to check my understanding of properties for use in solving an equation. Are the following true, for A, B, C as elements of $SE(3)$? $$ \log(ABC) ...
-1
votes
1answer
44 views

Solving a logarithmic inequality

For what values of $x$ it holds that: $$-\log_{e}(1-x) \geq x$$ and how can we prove this? $\log_{e}$ is the natural logarithm $\ln$.
0
votes
0answers
16 views

Fisher Transform: Use of the natural logarithm of negative number. How is it possible?

I have the following equation, from http://www.mesasoftware.com/Papers/USING%20THE%20FISHER%20TRANSFORM.pdf Because the parameters inside the log are (1+x)/(1-x), the output is always negative when ...
3
votes
2answers
92 views

Find $\frac{1}{\log 2}+\frac{1}{(\log 2)(\log 3)}+\frac{1}{(\log 2)(\log 3)(\log 4)}+ \cdots$

Is it possible to calculate the sum of $\dfrac{1}{\log 2}+\dfrac{1}{(\log 2)(\log 3)}+\dfrac{1}{(\log 2)(\log 3)(\log 4)}+ \cdots$?
0
votes
1answer
71 views

solving mod equation

I am attempting to solve $r_1$ in this equation: $$m + xr \equiv m_1 + xr_1 \pmod q$$ This is what I derived at: $$m-m_1 + xr / x \equiv r_1 \pmod q$$ I proceed to sub these with the necessary ...
0
votes
1answer
47 views

Maximize ratio of logarithms

How can one maximize the ratio of two logarithms $ \frac{\log{f(x)}}{\log{g(x)}}$ where the argument to each logarithm is the (positive) ratio of two first-degree polynomials? I have tried ...
3
votes
2answers
104 views

Proof of a closed form of $\int_0^1(-\ln x)^ndx$

$$\int_0^1(-\ln x)^ndx$$ Is there a step-by-step solution to a closed form of this expression? I've tried using different representations to re-write the expression but I couldn't find anything I knew ...
2
votes
3answers
126 views

Show without differentiation that $\frac {\ln{n}}{\sqrt{n+1}}$ is decreasing

Show that the function $\displaystyle \frac {\ln{n}}{\sqrt{n+1}}$ is decreasing from some $n_0$ My try: $\displaystyle a_{n+1}=\frac{\ln{(n+1)}}{\sqrt{n+2}}\le ...
2
votes
2answers
82 views

Show without derivative that function $\frac{\ln{n}}{ n\ln{\ln{n}}}$ is decreasing

I have a problem with showing the function $\displaystyle \frac{\ln{n}}{n \ln{\ln{n}}}$ is decreasing. I came to form $(n+1)^{\ln{\ln{n}}}<(n)^{\ln{\ln{(n+1)}}}$ and I don't know how to show that ...
2
votes
2answers
129 views

Solving the equation $\ln(x)=-x$

I tried solving this equation for a long time but did not succeed. Any help is appreciated. $$\ln x=-x$$ I am not sure the tag is correct, I am not familiar with English mathematical terms. Please ...
6
votes
1answer
107 views

$\log^2 (x^2) + \log (x-1) = 0$

I'm trying to solve the equation $\log^2 (x^2) + \log (x-1) = 0$ but all I could do is to show that $1 < x < 2$. Wolfram Alpha says that $x = 1.508554...$, this is good, but I really want to ...
2
votes
2answers
56 views

Upperbound this difference between two log expressions

I have the difference between the following log expressions and I am trying to bound the difference, $$F= \log \left(1+ \left(2+\frac{1}{\sqrt{2}}\right)^2 x^2\right) - \log \left(1+ ...
0
votes
0answers
35 views

Transforming a logarithmic expression?

Do you know any nice way to rewrite $\log(1-e^{A})$?
3
votes
8answers
298 views

Solution of an exponential equation

Probably very simple question. Why the solution of $$1=n(1-a)^{t}$$ in terms of $t$ is equal to: $$t=\frac{\ln n}{\ln \frac{1}{1-a}}$$
1
vote
1answer
40 views

How do I prove this derivation?

I hope you can help me with this one because I seem to not quiet get a start here :/ Lets say we got a $b\in\mathbb{R}_{\gt 0}$ and a $y\in\mathbb{R}$ and we define $b^y:=\exp\left(\ln b \cdot ...
1
vote
2answers
69 views

Why does this inequality stand?

I want to ask something about: "Since $i \log_e i$ is concave upwards, it is easy to show that $$\sum_{i=2}^{n-1} i \log_e i \leq \int_2^n x \log_e x \,dx \leq \frac{n^2 \log_e ...
5
votes
1answer
48 views

How to prove that $f(x) - f(x-1)$ approaches $\frac{\log_{10}(10)}{\log_{10}(e)}$?

Let $$f(x) = \sum_{n=1}^{10^x}\frac{1}{n}$$ I noticed that as x approaches $\infty$, $f(x) - f(x - 1) \approx 2.3025$. After a bit of experimenting, I found that $2.3025... = ...
1
vote
2answers
100 views

How does $n < 2^n$ become $\log n < n$ by taking log of both sides?

How does $n < 2^n$ become $\log n < n$ by taking the log of both sides? I understand the left side but I do not understand the right side of the inequality. The once was $\log 2^n$ becomes $n$ ...