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Questions tagged [logarithms]

Questions related to real and complex logarithms.

2
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2answers
46 views

Proof that a sequence of positive numbers satisfies $P_n ^2 \leq P_{n-1}\,P_{n+1}$.

Let $(P_n)$ be a sequence such that $P_0 = 1$ and $P_n = (\log{(n+1)})^n$. I'm trying to prove that $P_n ^2 \leq P_{n-1}\,P_{n+1}$ for $n \geq 1$ using an induction argument. For $n=1$ we have that $...
3
votes
2answers
40 views

Show that the series $\sum_{n=3}^{\infty} n(\log n)(\log {\log n})^2 a_n$ diverges.

Suppose that the series $\sum_{n=1}^{\infty} a_n$ converges conditionally. Show that the series $\sum_{n=3}^{\infty} n(\log n)(\log {\log n})^2 a_n$ diverges. Any hint on how to proceed?
0
votes
0answers
48 views

I have some problem with this logarithms , can i get a help? [on hold]

If $\log_{4} a = \log_{10} b =\log_{25} (a+b)$, what is $\frac ba$? A. $\frac{1+\sqrt5}{2}$ B. $\frac{1+\sqrt3}{2}$ C. $\frac45$ D. $\frac{25}{10}$ E. $\frac83$
1
vote
3answers
61 views

Solve $\log_a(\log_a x^n)$ if $n=a^2$ ; $x= e^2$

My brother challenged me to solve this problem. Trying since 2 days. I came up with $a^{a^y}= x^n$ assuming $y$ is $\log_a(\log_a x^n)$. There's no solution available on net as well. If someone can ...
1
vote
2answers
32 views

Solving for a variable in a complex algebra equation with logs and powers

I have a follow-up question to my last one. I need to solve for b for the following: $$d=\frac{s(\ln(o))^b}{s(\ln(g))^b}$$ Once again, this is beyond the level of ...
-2
votes
0answers
43 views

Looking For Advanced Logarithm Questions

do you guys know any site/contests that has advanced logarithm questions (that is grade 12 level)?? I'm doing the Euclid contest and I want to do more logarithm related questions..! Thank you!
4
votes
2answers
58 views

How to show $\int_{0}^{\infty}\frac{x}{x^{2}+1}\log\left(\left|\frac{x^r+1}{x^r-1}\right|\right)dx = \frac{\pi^2}{4r}$?

After playing around with a few values of r, I have the following conjecture: $$\int_{0}^{\infty}\frac{x}{x^{2}+1}\log\left(\left|\frac{x^r+1}{x^r-1}\right|\right)dx = \frac{\pi^2}{4r}$$ for r a real ...
2
votes
3answers
102 views

What is the function $y = (1+1/x)^x$ solved for $x$?

I came across this function in algebra ($e$ being its limit as $x$ goes to infinity) while studying compounded interest. Since this function is a little modified from the real interest formula $y=(1+1/...
2
votes
3answers
50 views

Properties of Logarithm (what is wrong with my math?)

The question is to find the intersection point between $y = log_22x$ and $y = log_4x$ So my first instinct was to make the same base 2. (Since 4 = 2 + 2) $log_22x = log_4x$ $log_22 + log_2x = ...
0
votes
1answer
29 views

Solving for a variable in an algebra equation with logs and powers

I'm working on a software project and I need to solve for b in the following equation: $$e^{\ln(gx)ab}-e^{\ln(x)ab}=c$$ I've tried a couple of online algebra ...
0
votes
2answers
46 views

Is there a characterization of entire functions with image $\Bbb C \setminus \{0\}$? [on hold]

Do we have a characterisation of entire functions with image $\Bbb C \setminus \{0\}$? If not, is there an example of such a function that is not in the form of $\exp(g)$ for some entire function $g$?...
0
votes
3answers
62 views

The derivative of $\ln(x)$ [on hold]

How can one prove the following by elementary means? $$\ln(x)'=\frac{1}{x}$$ Say we know that $$e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}.$$
0
votes
0answers
38 views

A quotient rule for logarithm in complex space?

I have the following formula: $$\frac{\ln(\frac{a}{x})}{k} + \frac{\ln(b) - \ln(x)}{k}$$ where $k>0$, $a>0$, but $b<0$ and $x<0$. I cannot use the quotient rule for logarithms to change ...
0
votes
0answers
25 views

Zeroes of a function with logarithms and peculiar symmetry

I encountered a strange function, defined on an open three-dimensional unit cube centered on the origin and taking values in $\mathbb{R}$. The function has some symmetry properties. It is not ...
0
votes
0answers
27 views

Finding the time complexity of the given code snippet.

Question Find the time complexity of the following code snippet ...
2
votes
5answers
62 views

Prove that $\log_n(n+1)\geq\log_{n+1}(n+2)$ for $n>1$

Prove that $\log_n(n+1)\geq\log_{n+1}(n+2)$ for $n>1$. So far I only know that $\log_n(n+1)>\frac{\log_n(n+2)}{\log_n(n+1)}$ Since $n>1$, LHS must be greater than RHS. Is there any other ...
-2
votes
1answer
60 views

What is the value of 1/log2+1/log3+1/log4+…+1/ log N? [closed]

How to calculate $1/\log2+1/\log3+1/\log4+...+1/ \log N?$ I have no idea why.
0
votes
3answers
47 views

When is the constant C part of the function when integrating

Lets take the following example: $\int \frac{x-4}{(x-2)(x-3)}dx$ The result I get is $2\ln|x-2|-\ln|x-3|+C$ The result in my testbook is: $\ln \frac{C(x-2)^2}{x-3}$ As $\ln \frac{C(x-2)^2}{x-3} = ...
2
votes
2answers
110 views

Inverse of $f(x) = \frac{x}{\log(x)}$

$\pi(x)$ is defined as number of primes less than or equal to x. Gauss showed that $\pi(x) \approx \frac{x}{log(x)}$. I want to calculate the inverse of that approximation because I want to estimate ...
2
votes
2answers
76 views

What algorithm do scientific calculators use to calculate Logarithms

I have been introduced to numerical analysis and have been researching quite a bit on its applications recently. One specific application would be the scientific calculator. From the information ...
0
votes
0answers
30 views

Can this function's positivity $\forall t \gt 5.56..$ be criteria for the Riemann hypothesis?

Let $Z \left( t \right) ={{\rm e}^{i \left( -i/2 \left( {\it ln\Gamma} \left( 1/4+i/2t \right) -{\it ln \Gamma} \left( 1/4-i/2t \right) \right) -1/2\,\ln \left( \pi \right) t \right) }}\zeta \...
1
vote
2answers
21 views

Find the range of a logarithmic function whose domain is all real numbers between 2 and 10 (exclusive).

$$h(x)=\log_{10}(x+1).$$ Shouldn't the range be all real numbers between $0.602$ and $1$ (inclusive)?
0
votes
1answer
37 views

Derivatives and functions and maxima

Let $$f(x)=\left\{% \begin{array}{ll} x^3-x^2+10x-5,& x\leq 1 \\ -2x+\log_2 (b^2-2),& x>1 \\ \end{array}% \right.$$ Find all possible real values of $b$ such that $f(x)$ has the ...
3
votes
2answers
142 views

Inequality with $abc=2$

Let $a,b,c$ be positive real numbers such that $abc=2$. Prove that $\frac{a+b+c}{4}\geq \sqrt[4]{\frac{a^2+b^2+c^2}{6}}$ Sorry, I don't have an idea for this problem.If I had, I would have showed them ...
1
vote
4answers
82 views

Is it true that $\ln(x)+\frac{1}{\ln(x)}>\ln(x+1)$ for $x>m$

I was wondering if it is true for $x>m$ where $m$ is a constant, we have: $\ln(x)+\frac{1}{\ln x} >\ln(x+1)$ If we plot the figure $\ln(​x)(\ln(​x+​1)-​\ln(​x))$ in google we see: As you can ...
1
vote
2answers
44 views

Expanding log problem

I found this site with online problems and answers. https://courses.lumenlearning.com/waymakercollegealgebra/chapter/expand-and-condense-logarithms/ I've tried several problems and my answer is ...
3
votes
3answers
145 views

Is there a clean way to show $\frac{\log(x-\tfrac12\log x)}{\log(x+\tfrac12\log x)}>1-\tfrac1x$ for all $x>1$?

In writing a paper I had to show that the inequality holds for large enough $x$, which is easy, but I ended up being pretty sure it holds for all $x>1$, so I would like to include the proof of the ...
2
votes
2answers
50 views

What is the relationship between modulo and $\log_2$ in this problem?

Here's a screenshot of a diagram from my computer architecture course. You don't need to understand the technical terms here. It's essentially mapping locations from one piece of memory to locations ...
0
votes
3answers
53 views

Estimating the value of $\ln2$ using $e^3$ and $2^{10}$

I found this question in an old MAT paper but I'm not getting very far. You are given that $e^3$ is approximately $20$ and that $2^{10}$ is approximately $1000$. Using this information a student can ...
1
vote
2answers
75 views

Why does $f(xy) = f(x)+ f(y) \implies f(x) = k \ln x$

My textbook says $f(xy) = f(x)+ f(y) \implies f(x) = k \ln x$ But if I'm not mistaken, $f(x)$ could be a logarithmic function with any base? So why only $\ln x$?
0
votes
1answer
36 views

Need to solve equations which take vector inputs. I have tried with matlab solve() but I'm getting zeros for both tau_new & b

rc=1; x=linspace(0,1,1000); R=exp(1).*x.*rc; b0=.3614/sqrt(2); eqn1=(b./(8*pi)).(1./R).(log(R./rc)+1)+(b./(2*b0))-tau_new==0, eqn2=(b./(2*pi)).(1./R).(log(R./rc))+(b./(b0))-tau_new==0]; vars=[...
8
votes
1answer
135 views

ln(2) contradiction

$\ln2\approx.693$, according to my calculator. It can be written as the infinite sum $$1-\frac12+\frac13-\frac14+\frac15-\frac16+\frac17-\frac18+\frac19-\frac1{10}\dots$$ Rearranging this infinite ...
1
vote
0answers
50 views

Finding argmax of an integral — can I take logarithm of one of the terms?

Suppose I need to solve for the argmax of $x$ for an integral that can be factored as follows: $$ A = \mathrm{argmax}_x \int_Y f(x,y)g(y)dy $$ Where both $f()$ and $g()$ are strictly positive, and ...
-1
votes
1answer
52 views

logarithms: $a^n =b$, got $\ln(x)$, but how do I get “$n$” using it?

I learned a fast way to get $x$ in $\ln(x)$ using ($n= x-1$ btw) $$\ln(x)=2\left(\frac{1}{2n+1}+\frac{1}{3(2n+1)^3}+\frac{1}{5(2n+1)^5}+\frac{1}{7(2n+1)^7}+\frac{1}{9(2n+1)^9}...\right) $$ from wiki,...
2
votes
3answers
402 views

Log inequalities

Solve the inequality $\log(5^{1/x} +5^3) < \log 6 + \log 5^{1+\frac{1}{2x}}$ I came up with an equation $5^{\frac{4x-1}{2x}} + 5^{\frac{-2x+1}{2x}}-6<0$ Which I couldn't solve, i tried to ...
1
vote
2answers
80 views

$\arg(\overline{z}), \arg(z^2)$ [closed]

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 3.42,43 Exer 3.42 multiple-valued: NO in general. $$\arg(1+i)=\frac \pi 4 + 2k \pi$$ $$...
0
votes
1answer
28 views

Determine whether the given series is absolutely convergent or conditionally convergent

Consider the series $$\sum_{n=1}^\infty \log\left(1+\frac{1}{|\sin(n)|}\right).$$ Determine whether it converges absolutely or conditionally. I am trying to apply Cauchy condensation test, but I ...
0
votes
0answers
51 views

How to find log of “sum of two matrices”?

I want to find log (A + B ) where A and B are matrices. The context is that I want to find the Von Neumann entropy which is given by: $Entropy = - Trace [\rho log (\rho) ]$ where $\rho$ is a matrix....
-1
votes
0answers
32 views

Sum of series 1/log(n/(2^k)) [duplicate]

What will be the sum of the given series? $\frac{1}{\log_2 n} + \frac{1}{\log_2 (n/2)} + \frac{1}{\log_2 (n/4)} + … + \frac{1}{\log_2 (n/(2^k)} $ where $k = \log_2 n - 1$
-1
votes
3answers
77 views

Can one solve for $x$ in the equation $8^x=16x$ [closed]

The question is whether or not it is possible to solve for $x$ in the equation $8^x=16x$. [![enter image description here][1]][1] [1]: https://i.stack.imgur.com/ypio1.jpg any assistance will be ...
0
votes
2answers
141 views

The definition of $\ln(x)$

When I taught my student the logarithm, he asked me about the historical definition of $\ln(x)$. The first definition I found is that $$\ln(x)=\int_{1}^{x}{ \frac{dt}{t} } $$ Defined as the ...
0
votes
2answers
34 views

Questions about Newman's simplified proof of Ramanujan's partition formula

Recently I started to go through Newman's proof of Ramanujan's asymptotic formula for the number of partitions $p(n)$. I got stuck right in the beginning, where we have $f(z) = \prod_{n=1}^\infty \...
2
votes
1answer
31 views

Exchanging max, log and absolute value

Let $N \in \mathbb{N}$, $f,g: [N] \to (0,1]$. It is a consequence of the triangle inequality that : $$\left|\max_{n \in [N]} f(n) - \max_{n \in [N]} g(n)\right| \leq \max_{n \in [N]} \left| f(n) - g(...
1
vote
1answer
48 views

How do I solve this logarithm problem? [closed]

I'm trying to solve this problem: If $\log_{27}(a)=b$, find $\log_{\sqrt[6]{a}}\sqrt{3}$ However, I'm unable to see any connection in those given information. How can I solve this logarithm?
-2
votes
1answer
30 views

the ln of a variable

I have the equation $2t-5.682511886=-2*\ln(3t)+\ln(30t)$ I need to solve this equation to find t but I am unable to do that because of the natural logs So my question is how do you deal with ...
5
votes
5answers
118 views

Solve: $2^x\Bigl(2^x-1\Bigl) + 2^{x-1}\Bigl(2^{x-1} -1 \Bigl) + … + 2^{x-99}\Bigl(2^{x-99} - 1\Bigl) = 0$

The question says to find the value of $x$ if, $$2^x\Bigl(2^x-1\Bigl) + 2^{x-1}\Bigl(2^{x-1} -1 \Bigl) + .... + 2^{x-99}\Bigl(2^{x-99} - 1 \Bigl)= 0$$ My approach: I rewrote the expression as, $$2^x\...
-1
votes
1answer
21 views

Domain and range of a logarithmic absolute value function.

$$y = \ln (|\ln x|)$$ Attempt at solution: For the function to be defined, $|\ln x|> 0$ $\implies ln x > 0 $ $\implies x > 1$ [$\because$ anti-logging both sides.] But clearly, the ...
1
vote
2answers
65 views

Find $y = \dots$ in $x = \frac{\ln{y}+1}{\ln{y}-1}$

The problem asks to get y out of x. Like this: $$x=\log{y}\implies y=e^x$$ So I have this: $$x = \frac{\ln{y}+1}{\ln{y}-1}$$ This is what I've attempted so far: $$x = \frac{\ln{y}+1}{\ln{y}-1} \...
3
votes
2answers
57 views

Logarithm equation

How to solve this logarithm equation? $$\frac12\cdot[\log(x) + \log(2)) + \log[\sqrt{2x} + 1] = \log(6).$$ The answer is $2$. I've tried to solve it, but I don't know how to proceed: $\frac12\log(...
0
votes
2answers
30 views

Derivative of Composite Natural Logarithm

What is the derivative of $\ln\left(\left(\frac{1-x}{1+x}\right)^2\right)$ with respect to $x$? I used chain rule and my answer was $-4\frac{1-x}{1+x}$. But the answer should be $\frac{-4}{1-x^2}$. Is ...