Questions related to real and complex logarithms.

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2
votes
3answers
118 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
36 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
34 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 ...
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
34 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 $...
1
vote
7answers
96 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
81 views

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

Is $\log_27$ a rational number?
2
votes
4answers
68 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
32 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
20 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
55 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
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
3answers
20 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
45 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
74 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
58 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
64 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 &...
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
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.
0
votes
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 \...
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
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 ...
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 ...
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
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 - \...
-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

Solve logarithm without calculator for exam practice [closed]

How to solve $$\log_{3}(x)+\log_{3}(3x) = 3$$ ?
0
votes
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 $$\...
0
votes
2answers
62 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
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 $$\frac{-217}2−20\,\ln(2)−173\,\ln(11)+732\,{\rm arctanh}\left(\frac58\right)?$$ ...
1
vote
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 ...
0
votes
2answers
38 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 ...
1
vote
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? ...
1
vote
0answers
28 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
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 ...
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 ...
0
votes
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)...
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)^...
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 $x=10^{5/2}$...
0
votes
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 ...
1
vote
2answers
63 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, ...
12
votes
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 ...
4
votes
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^...
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
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 $x>2$.
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 ...
-1
votes
5answers
72 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 $$ \...