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Questions related to real and complex logarithms.

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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: $2e^x -1 &... 1answer 31 views Convergence of a sequence of roots of continous functions Let$(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 ... 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 ... 2answers 46 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. 0answers 65 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$n. And further that the following inequality holds: \begin{align} \sum_i a_i \leq \... 2answers 30 views Show that xy=100. Given2\log x^3y=6+3\log y-\log x$. 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$x=y$. 0answers 24 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 ... 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 ... 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 ... 3answers 43 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 $$2\ln8 - \... 2answers 72 views Proof without words for logarithmic funtions [closed] I'm looking for a PROOF WITHOUT words for logarithms. The only one I've seen is calculus based and I need one for a younger audience. Any help/suggestions would be appreciated! This is the example I ... 1answer 14 views forming log equation from graph points Okay so I need to form a logarithmic equation from the points (1960,4.7) (1964,5.1) (1968,5.4). I have 'guess and checked' to get the equation 2.7421 log(x-1950)+1.9579, and was wondering if there was ... 4answers 49 views Solve logarithm without calculator for exam practice [closed] How to solve$$\log_{3}(x)+\log_{3}(3x) = 3$$? 2answers 47 views Evaluation of x in \log_{\frac{3}{4}}\left(\frac{x}{3}\right)+\log_{\frac{1}{2}}\left(\frac{x}{2}\right) = -2 Evaluation of x in$$\log_{\frac{3}{4}}\left(\frac{x}{3}\right)+\log_{\frac{1}{2}}\left(\frac{x}{2}\right) = -2$$\bf{My\; Try::} Here x>0\;, Now Using Properties of \log\;, We get$$\... 2answers 61 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'... 1answer 68 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 $$\frac{-217}2−20\,\ln(2)−173\,\ln(11)+732\,{\rm arctanh}\left(\frac58\right)?$$ ... 2answers 23 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 ... 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 ... 1answer 43 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? ... 0answers 27 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 ... 1answer 29 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 ... 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 ... 2answers 33 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)... 0answers 72 views Did I compute this expression with logarithm correctly? Let$2\leq e\leq r$and $$a_{n,k}=2(e+1)^{2(n+k)-1},$$ $$b_{n,k}=2\cdot[ e+e^2+e^3+e^3(e+1)+e^3(e+1)^2+\ldots +e^3(e+1)^{2(n+k)-4}],$$ $$c_{n,k}=2\cdot [2(e+r)e(e+1)^{2(n+k)-4}+(2(n+k)-4)(e+r)(e+1)^... 2answers 2k views Why do these “equal” logarithms give different answers This came across a discussion amongst Algebra 2 teachers at my school. We know a\log x= \log x^a Say 2\log x=5 \log x^2 =5 When \log x=\log_{10} x Solving the first equation yields x=10^{5/2}... 2answers 47 views Finding the value of x, logarithms and exponentials I'm working through some logs and exponentials questions at the moment in order so that I might be a little prepared for any I might utilize in a science PHD. I'm currently getting through the ... 2answers 62 views What is the difference in this question between \log and \lg? Am I right in assuming that \lg just refers to \log base (10)? Whereas \log is just any unspecified log? I'm solving \lg{15}-\lg{5} Am I good to just use the standard rules of logarithms, ... 3answers 869 views Integral involving logarithm: \int_0^\infty \frac{ \ln x}{(x+a)(x+b)} dx How to solve the following integral$$\int_{0}^{\infty} \frac{ \ln x}{(x+a)(x+b)} dx,$$where a,b>0 and a \neq b. I was looking for some kind of substitution. However, I don't see an obvious ... 3answers 158 views 2^n=n and similar equations Is it possible to solve equations in the form k^n=n for n and if so, How? I am new to logarithms and so would be glad if someone could explain even if there is an obvious answer. Also What about k^... 4answers 25 views Half-life of Am-241, 3 micrograms decays over 9 years, how much if left? 3 micrograms of Americium-241, which has a half life of 432 years. After 9 years how much will remain? I'm not sure of the formula to use or how to calculate it. I'm assuming it's exponential ... 5answers 97 views Prove \ln^2(x)>\ln(x+1)\cdot\ln(x-1) for x>2 [closed] Could anybody please help prove the following: \ln^2(x)>\ln(x+1)\cdot\ln(x-1), for x>2. 1answer 50 views How to find exact length of digits or number of digits of a^b? If a and b are positive integer then what is length of digits of a^b? I have worked so far and formula works fine. To find the exact length of digits of a^b where a\gt 0, b\gt 0: Number ... 5answers 71 views Squaring a logarithm when the base is a square root How is this equality obtained?$$ -\log_\frac 1 {\sqrt 2}(x - 7) = \log_2 (x - 7)^2 $$I understand the process until this point$$ \log _\sqrt 2 (x-7) . $$How do I get from there to$$ \... 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 ... 1answer 36 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 ... 0answers 33 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. ... 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 ... 1answer 46 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{\...
$$a^{bx} = c$$ Solve for x $$\log a^{bx} = \log c$$ $$bx \log a = \log c$$ $$x = \frac{\log c}{b \log a}$$ Is this correct? Thanks :)