Questions related to real and complex logarithms.

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8
votes
6answers
3k views

An alternative way to calculate $\log(x)$

How can I replace the $\log(x)$ function by simple math operators like $+,-,\div$, and $\times$? I am writing a computer code and I must use $\log(x)$ in it. However, the technology I am using does ...
2
votes
1answer
42 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
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 ...
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
1answer
62 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: $2e^x -1 &...
-4
votes
2answers
44 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
64 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 \...
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 ...
0
votes
2answers
30 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
23 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
31 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 ...
1
vote
2answers
61 views

What is the difference in this question between $\log$ and $\lg$?

Am I right in assuming that $\lg$ just refers to $\log$ base ($10$)? Whereas $\log$ is just any unspecified log? I'm solving $\lg{15}-\lg{5}$ Am I good to just use the standard rules of logarithms, ...
0
votes
3answers
72 views

Integrate a power of logarithm [closed]

Is there some way, how to solve this problem? $$ \int \ln^n(x) dx \text{, where } n \in \mathbb{N} $$ I really don't know, what to do with $n$.
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
40 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
72 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
49 views
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
60 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 doesn'...
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
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 ...
1
vote
2answers
89 views

How to solve $3^{\sqrt{\log_{3}{x}}}+x^{\sqrt{\log_{3}{x}}}=6$

How can i solve the following equation? $$ 3^{\sqrt{\log_{3}{x}}}+x^{\sqrt{\log_{3}{x}}}=6 $$ It is clear that $x=3$ is a solution of this equation. But how can i prove that there is another solution ...
0
votes
1answer
41 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
25 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 ...
4
votes
3answers
140 views

$2^n=n$ and similar equations

Is it possible to solve equations in the form $k^n=n$ for n and if so, How? I am new to logarithms and so would be glad if someone could explain even if there is an obvious answer. Also What about $k^...
0
votes
0answers
72 views

Did I compute this expression with logarithm correctly?

Let $2\leq e\leq r$ and $$a_{n,k}=2(e+1)^{2(n+k)-1},$$ $$b_{n,k}=2\cdot[ e+e^2+e^3+e^3(e+1)+e^3(e+1)^2+\ldots +e^3(e+1)^{2(n+k)-4}],$$ $$c_{n,k}=2\cdot [2(e+r)e(e+1)^{2(n+k)-4}+(2(n+k)-4)(e+r)(e+1)^...
2
votes
1answer
34 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 ...
-1
votes
5answers
68 views

Squaring a logarithm when the base is a square root

How is this equality obtained? $$ -\log_\frac 1 {\sqrt 2}(x - 7) = \log_2 (x - 7)^2 $$ I understand the process until this point $$ \log _\sqrt 2 (x-7) . $$ How do I get from there to $$ \...
0
votes
2answers
32 views

Problem solving with mass in terms of logs and exponentials

I've just been accepted to take my PHD in chemical engineering in Melbourne next year. Some how I have gone from the age of 17 with out taking too many extra maths classes and so at the moment (I'm 26)...
1
vote
2answers
101 views

Avoiding subtraction for finite difference with log and exp

I want to approximate the derivative of f(x) Finite difference $f'(x) \approx \frac{f(x+h)-f(x)}{h}$ I was taught that the error from the subtraction is blown up for small h. This I can verify with ...
12
votes
3answers
867 views

Integral involving logarithm: $\int_0^\infty \frac{ \ln x}{(x+a)(x+b)} dx$

How to solve the following integral $$\int_{0}^{\infty} \frac{ \ln x}{(x+a)(x+b)} dx,$$ where $a,b>0$ and $a \neq b$. I was looking for some kind of substitution. However, I don't see an obvious ...
9
votes
2answers
2k views

Why do these “equal” logarithms give different answers

This came across a discussion amongst Algebra 2 teachers at my school. We know $a\log x= \log x^a$ Say $2\log x=5$ $\log x^2 =5$ When $\log x=\log_{10} x$ Solving the first equation yields $x=10^{5/2}$...
1
vote
1answer
50 views

How to find exact length of digits or number of digits of $a^b$?

If $a$ and $b$ are positive integer then what is length of digits of $a^b$? I have worked so far and formula works fine. To find the exact length of digits of $a^b$ where $a\gt 0, b\gt 0$: Number ...
0
votes
2answers
44 views

Finding the value of x, logarithms and exponentials

I'm working through some logs and exponentials questions at the moment in order so that I might be a little prepared for any I might utilize in a science PHD. I'm currently getting through the ...
1
vote
4answers
25 views

Half-life of Am-$241$, $3$ micrograms decays over $9$ years, how much if left?

$3$ micrograms of Americium-$241$, which has a half life of $432$ years. After $9$ years how much will remain? I'm not sure of the formula to use or how to calculate it. I'm assuming it's exponential ...
-2
votes
5answers
96 views

Prove $\ln^2(x)>\ln(x+1)\cdot\ln(x-1)$ for $x>2$ [closed]

Could anybody please help prove the following: $\ln^2(x)>\ln(x+1)\cdot\ln(x-1)$, for $x>2$.
-2
votes
1answer
39 views

Does, S = k ln W == W = e^s/k? [closed]

"Boltzmann's equation relates the entropy S of an ideal gas to the number W of microstates corresponding to a given macrostate, via the equation S = k ln W where k is the so-called Boltzmann ...
0
votes
1answer
35 views

Newton's Law of Cooling (and Heating)

The Formula for the equation is as follows: $$T(t)=\frac {\int^t(−T_s)ke^{-kt'}dt'+C}{e^{-kt}}$$ This formula is needed to determine the temperature at time $t$, $T(t)$, of an object as it begins to ...
1
vote
0answers
18 views

Interpolation / point fitting onto a logarithmic line segment

I have figure which is logarithmic scale on both axis. There's a line on that figure, I know two points on that line and want to interpolate a third point on that line based on the two known points. ...
1
vote
1answer
25 views

Newtons Law of Cooling in Forensic Science

Question goes: Law enforcement would like to know the time at which a person died. The investigator arrived on the scene at 8:15pm, which we will call $t$ hours after death. At 8:15 (i.e $t$ hours ...
0
votes
1answer
45 views

Logarithm Rules Ambiguity

I'm having some problems explaining myself the following ambiguity. According to logarithm rules: $\ln6=\ln(2\cdot3)=\color\red{\ln2+\ln3}$ $\ln6=\ln((-2)\cdot(-3))=\ln(-2)+\ln(-3)=\color\red{\...
0
votes
1answer
45 views

Logarithm problem question

$$a^{bx} = c$$ Solve for x $$\log a^{bx} = \log c$$ $$bx \log a = \log c$$ $$x = \frac{\log c}{b \log a}$$ Is this correct? Thanks :)
1
vote
5answers
68 views

Solve for $x$ : $\log_e(x^2-16)\lt \log_e(4x-11)$

$\log_e(x^2-16)\lt \log_e(4x-11)$ My attempt: Since the base is $\gt 1$, we have from the above , $$x^2-16-4x+11\lt 0\\ \implies x^2-4x-5\lt 0\\ \implies(x-5)\cdot (x+1)\lt 0$$ If I say $(x-5)\gt ...
2
votes
2answers
32 views

Questions about Exponentiation and roots and logarithms.

in this page a few questions I want to ask you about the Exponentiation and roots and logarithms: What and how the Exponentiation definition can be defined by real numbers.? What is the overall ...
0
votes
2answers
37 views

Why is my answer incorrect for this differentiation question?

$$y = x* ((x^2+1)^{1/2})$$ I must find $$dy/dx$$ $$u = x, v = (x^2+1)^{1/2}$$ To do this I must use the product rule and the chain rule. To get dv/dx, $$(dv/dx) = (1/2)*(b)^{-1/2}*2x $$ $$(dv/dx) ...
0
votes
1answer
29 views

derive the pdf for “difference of log-normal distributions”

Can someone please help me to derive pdf for $X$, $$ X = \frac{\ln(f_1) - \ln(f_2)}{b_2-b_1} $$ here $f_1$ and $f_2$ are normal distributions with different means and standard deviations, and $b_1$ ...
0
votes
1answer
24 views

Log power rule problem

According to many parts of the Internet, this log rule is used. log(a^b) = b*log(a) The proof is: Now let's say I want to use the rule in a Cartesian ...