Questions related to real and complex logarithms.

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0
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1answer
22 views

Exponent to maximize the expression $log_b \left(a\frac{b-1}{b^k-1}\right)$

Given $ a, b \in \mathbb N $, how to maximize the expression $$ log_b \left(a\frac{b-1}{b^k-1}\right) \in \mathbb N $$ Put differently, what is the minimum $k \in \mathbb N $ verifying $$ ...
1
vote
1answer
51 views

How to solve this equation using logarithms?

I have to solve for all real values of $x$. $(5+2\sqrt6)^{x^2-3}+(5-2\sqrt6)^{x^2-3}=10$ I tried to take $\log_{10}$ on both sides but could not do this. How do I do this?Thanks for any hint or ...
0
votes
2answers
38 views

Minimum of the antilogarithm

Given a $ a \in \mathbb N $, what is the lowest $ b \in \{1, ..., a \} $ for which $ log_b a \in \mathbb N $ ? How to compute this function in a non-iterative way? Examples (even if too obvious): $ a ...
-1
votes
3answers
45 views

Proof of a logarithm identity

I would like to know how to prove the following log identity: $x^{\frac{\log(\log(x))}{\log(x)}} = \log(x)$
2
votes
2answers
61 views

Proving that two branch cuts can cancel out

Define the following functions $\mathbb{C}\to\mathbb{C}:$ $$u(z)=\frac{\log \left(z+\frac{1}{2}\right)}{z}\quad \left[-\pi\leqslant\arg \left(z+\tfrac12\right)<\pi\right];\quad v(z)=\frac{\log ...
1
vote
1answer
25 views

During the 56th month or the 57th month?

A car depreciates in value according to the model $$V=Ak^t$$ where £$V$ is the value of the car $t$ months from when it was new. Its value when new was £$12499$ and $36$ months later its value was ...
-3
votes
1answer
47 views

What is the solution for the Logarithmic CompSci question? [on hold]

Facts: My alphabet contains m digits. Each digit is represented in memory by 1 byte. A word is any string of digits. Words are separated by a single non-alphabetic character. It is represented in ...
0
votes
2answers
44 views

Complex logarithms when computing real-valued integral

My question arise when I try to calculate real-valued integral, specifically, I want to evaluate the integral \begin{equation} \int_0^1 \frac{\ln \left(\frac{x^2}{2}-x+1\right)}{x} dx \end{equation} ...
0
votes
7answers
1k views

What is “8 log 2”?

When someone says "8 Log 2" what does this equate to in writing? Does it mean the following? $$ \log _{2} 8 $$ And if so, what is the value of this? When I plug those numbers into this log ...
2
votes
3answers
113 views

How to solve $3(a+1)(b+1)=3^a \times 2^b$?

Hi I'm new to logarithms and not sure how to solve equations involving logarithms. I managed to find this equation to answer a problem solving question, however now I do not know how to solve the ...
0
votes
2answers
24 views

Graphing log with number in front of “log”

When I have something like $y = log_2(x)$ I know that I have to turn it into exponential form and get: $2^y = x$. Next, I make a table for $X,Y$ and choose about 5 values for $y$, typically $-1, 0, 1, ...
1
vote
0answers
27 views

Order Size estimation of converging sum used for approximation of logarithm

I know it can be shown that $\log n=\sum_{i=1}^\infty \frac{(n-1)^i}{in^i}$ for $\forall n\in\Re$ where $n\ge1$ For given natural m, I tried to find the order size of k = f(m,n) in order for the ...
1
vote
0answers
21 views

Proof this limit superior is finite.

Let $\{ w_n \}$ be a sequence of non-negative numbers and put $M_n=\sum_{k=1}^n w_k^2 \xrightarrow{n\to\infty} \infty $. Proof that $$\limsup_{n\to\infty} \dfrac{\ln \ln \sqrt{M_n \ln \ln M_n} }{\ln ...
0
votes
2answers
34 views

Logs - changing the base to evaluate

Just a bit confused about how to evaluate the following $$\log_3 8\times \log_5 9\times \log_2 5$$ What I have done so far: I have used the change of base rule to change each log to base $3$, ...
0
votes
2answers
33 views

log to exponential form, but with number in front of log

So I understand how to put a log equation into exponential form. For example, $y = \log_2(x)$ is $2^y = x.$ However, I don't understand what to do when there is a number in front of $\log$, such as ...
-5
votes
2answers
69 views

How to proove that $\log(n)$ < $\sqrt{n}$? [closed]

How to prove that $\log(n)$ < $\sqrt{n}$ I understand that O($\log(n)$) should work faster than O($\sqrt{n}$) but I can't understand how?
1
vote
7answers
90 views

Solve the following equation : $\log_2(x)*\log_4(x)*\log_8(x)=4.5$

I have the following equation : $$\log_2(x)*\log_4(x)*\log_8(x)=4.5$$ Usually, I do post what I made to do, but in this case a friend of mine tackle me with this question after I didn't mess with ...
3
votes
2answers
80 views

Is the value of $\log_27$ a rational number?

Is $\log_27$ a rational number?
2
votes
4answers
63 views

Use the definition of $\ln(x)$ as an integral to show that $f(x)=\frac{ln(x)}{x^2} \leq 1/x$ for all $x\geq 1$.

As the title says, if we let $$f(x)=\frac{ln(x)}{x^2}$$ I know that $$ln(x)=\int_1^x \frac{dt}{t} dt$$ Since $x^2 >0$ we can rewrite the question as $ln(x) \lt x$ $\forall$ $x\ge1$ How do we ...
1
vote
1answer
31 views

Logarithmic Taylor series question [closed]

Consider the transformation of variables, x = $\frac{y+1}{y-1}$ How would you develop log(x) as a Taylor Series in y about zero?
0
votes
2answers
18 views

How to show that: $\log_a (x^{a}-x)-\log_a \Big(\dfrac{x^{a}-x}{a}\Big)=1$, where $a$ and $x$ are positive integers.

I was studying Fermat's Little Theorem and Logarithm to see if there is any interesting result or correlation exist between the two. So I came up with this equation. I know few basic logarithmic ...
2
votes
2answers
52 views

$\ln(x)$ and Big O notation

I have tried to assert that $\ln(x)=O(x^0)$ a few times, but it seems fairly obvious that this statement should be false, and so I've been faced with some rightful speculation. My reason is that ...
4
votes
1answer
32 views

Balanced partition of $\{\ln 3, \ln 4,\dots,\ln n\}$

For a positive integer $n\ge 3$, let $A_n=\{\ln 3, \ln 4,\dots,\ln n\}$. Does there exist $N$ such that for all $n>N$, the set $A_n$ can be partitioned into two sets so that their sums differ by no ...
2
votes
1answer
40 views

Is there a simple solution to these 2 equations without trying all possible values?

Now I have two equations and the computation is in a finite field GF(p), where p is a prime. $x, y$ are unknown, and $a, b$ are known. ($0<y<p-1$, and $0<ax<p-1.$) $\begin{cases} x^y ...
0
votes
3answers
18 views

Log rules being applied to LN (Silent Logs)

I am doing a question on logarithms and am a bit confused regarding a solution I have found. As you can see below in the solution at one point the questions requires you to square (4ln(2))^2. When I ...
0
votes
0answers
40 views

If $\log_510=\log_7x(\log_nm)$ then the values of x,m and n are?

I have the question that if $\log_510=\log_7x(\log_nm)$ then values of $x$,$m$ and $n$ are? This question looks easy but i tried to get the expression down to the form $$\log_ab=\log_ac\tag{1.}$$ and ...
2
votes
1answer
70 views

Conditional Expectation of the minimum of two identical log-normal distributions

I'd like to compute the closed form mean of the minimum of two truncated log-normal distribution (on another interval than its truncation). I have: $\int_{a}^{\infty} \int_{a}^{\infty} min(v, v') \ ...
0
votes
3answers
56 views

Solve $ \log_3x\log_4x(\log_5x-1)$=$\log_5x(\log_4x+\log_3x)$

Solve $ \log_3x\log_4x(\log_5x-1)$=$\log_5x(\log_4x+\log_3x)$ for $x>0$. The constants $3$, $4$ and $5$ are meant to be the bases of the logs.
0
votes
1answer
61 views

Find all real solutions of $ \frac{ae^x}{2e^x-1} < 1 $

Question: Find all real solutions of $ \frac{ae^x}{2e^x-1} < 1 $, where $a$ is a positive constant. This is what I have attempted: Consider $$ \frac{ae^x}{2e^x-1} < 1 $$ Case 1: ...
0
votes
1answer
31 views

Convergence of a sequence of roots of continous functions

Let $(f^n,n\in\mathbb{N})$ be a sequence of complex continous functions so that $f^n(u)\longrightarrow f(u)$ uniformly to a complex continous function f if $n \longrightarrow \infty$. I addition I ...
2
votes
2answers
34 views

Question on logarithm Exponentiation

I know it's not the best title but I had no idea how to be specific about it. Basically what I'm looking for is a rule that states how $$\log^2(a^{f(x)})$$ works. Does it become $$f(x)\log^2(a)$$ or ...
-4
votes
2answers
42 views

Find the product of $\log_{2005}(1/2)\log_{2004}(1/3)\log_{2003}(1/4)\cdots\log_2(1/2005)$. The bases are $2005,2004,2003,\ldots,2$ [closed]

This question was answered in this site itself by Mark Bennet. But I didn't understand how the logs got cancelled out.
0
votes
0answers
63 views

Linear Inequality Implies Log Inequality

Imagine I have three sets of strictly positive real numbers: $a_i,b_i,c_i>0$, $\forall i=1,\ldots,n$. For finite $n$. And further that the following inequality holds: \begin{align} \sum_i a_i \leq ...
0
votes
2answers
28 views

Show that xy=100. Given $2\log x^3y=6+3\log y-\log x$.

Given $2\log x^3y=6+3\log y-\log x$, x and y are positive integers. Show that $xy=100$. I have tried until $x^7=10^6 y$. Now, my problem is how to prove $x=y$.
0
votes
0answers
19 views

Laurent series of logarithm

Lets have a function $$f(z)=\ln(\frac{z-a}{z-b})$$ on the region where it is holomorphic(off course). I want to find the laurent series for this function. Now finding the taylor expansion of this ...
0
votes
2answers
29 views

Calculating the mass xkg of radio-active substance pertaining to days after starting timing

Just testing myself with some tricky questions in my further maths textbook. This one states that the mass xkg of a radio-active substance remaining in a sample t days after starting timing is given ...
0
votes
1answer
29 views

Suppose there is $log_{a}^{*}x$ and $\log_{b}^{*}x$ then $\log_{a}^{*}x = O(\log_{b}^{*}x)$

Consider two $a,b \in R$. So my question is : Suppose there is exist $log_{a}^{*}x$ and $\log_{b}^{*}x$ then $\log_{a}^{*}x = O(\log_{b}^{*}x)$ NB: $\log^{*}{n} = 1+\log^{*}{\log{n}}$ Actually I ...
-1
votes
3answers
39 views

Simplifying, using logarithmic laws

I'm just going through some simplifying questions in my textbook. It asks me to simplify a series of expressions. I'm fine with the logs and lgs, but I'm struggling on this one: Simplify $$2\ln8 - ...
-1
votes
2answers
71 views

Proof without words for logarithmic funtions [closed]

I'm looking for a PROOF WITHOUT words for logarithms. The only one I've seen is calculus based and I need one for a younger audience. Any help/suggestions would be appreciated! This is the example I ...
0
votes
1answer
14 views

forming log equation from graph points

Okay so I need to form a logarithmic equation from the points (1960,4.7) (1964,5.1) (1968,5.4). I have 'guess and checked' to get the equation 2.7421 log(x-1950)+1.9579, and was wondering if there was ...
-3
votes
4answers
48 views

Solve logarithm without calculator for exam practice [closed]

How to solve $$\log_{3}(x)+\log_{3}(3x) = 3$$ ?
0
votes
2answers
46 views

Evaluation of $x$ in $\log_{\frac{3}{4}}\left(\frac{x}{3}\right)+\log_{\frac{1}{2}}\left(\frac{x}{2}\right) = -2$

Evaluation of $x$ in $$\log_{\frac{3}{4}}\left(\frac{x}{3}\right)+\log_{\frac{1}{2}}\left(\frac{x}{2}\right) = -2$$ $\bf{My\; Try::}$ Here $x>0\;,$ Now Using Properties of $\log\;,$ We get ...
0
votes
2answers
57 views

Show that $\sum _{k=1} ^N \frac 1 {\sqrt {k^2 + 1} + k} > \frac 1 2 \ln \frac {2N+1} 3$, where $N$ is natural number.

Show that for $N = 1,2,3,\dots$ we have $$\sum _{k=1} ^N \frac 1 {\sqrt {k^2 + 1} + k} > \frac 1 2 \ln \frac {2N+1} 3$$ I got this as a calculus homework. I am supposed to show this, but it ...
0
votes
1answer
67 views

can someone explain this simplification for me?? [closed]

Can someone tell me how $$−56−173\,\ln(11)+366\,\ln(13)−\left(\frac{105}2+20\,\ln(2)+366\,\ln(3)\right)$$ simplifies to $$\frac{-217}2−20\,\ln(2)−173\,\ln(11)+732\,{\rm arctanh}\left(\frac58\right)?$$ ...
1
vote
2answers
22 views

Logarithmic square

I can't understand if there is any such formula for $(\log_{b}a)^2$. Are there any? $\log_{b}(a^2) = 2\log_{b}{a}$ But if the whole log is squared is there any such formula or the same formula is ...
0
votes
2answers
37 views

Ascertaining a from logarithmic equations

I've just been accepted on to a PHD program at Melbourne, studying chemical engineering. I'm working my way through some standard pure and further mathematics books just to get the concepts into my ...
0
votes
1answer
40 views

Solve for x: $4\log_{x/2}(\sqrt{x}) + 2 \log_{4x} (x^2) = 3 \log_{2x} (x^3)$

$$4\log_{x/2}(\sqrt{x}) + 2 \log_{4x} (x^2) = 3 \log_{2x} (x^3)$$ This is a different type of equation. Our school has not taught this type yet. But this came in our exams. Can someone please help? ...
1
vote
0answers
19 views

Pollard Rho - DLP Algorithm Implementation

I am working with Pollard Rho Algorithm DLP. I have developed in Java and Python the way to calculate the table to find the collisions, and then using congruences and some others tricks I am getting ...
0
votes
1answer
28 views

Finding the solution of logs and exponentials equations to 2 decimal places

I'm going through maths textbooks at a rather fast pace at the moment as I have been accepted to take my chemical engineering PHD in Melbourne next year. I have been doing really well at the log ...
2
votes
1answer
29 views

Maximizing sum of logarithms (Z-channel capacity)

In the context of information theory, I am trying to maximize the following function (mutual information of the Z-channel's input and output) with respect to $p$ in order to derive Z-channel's ...