Questions related to real and complex logarithms.

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Domain of definition of the function

I was going through some questions of Relations and Functions and now I am stuck to one. Question says Question: Domain of definition of the function $$f(x)=\frac{9}{9-x^2}+\log_{10}(x^3-x)$$ ...
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104 views

A logarithm integral

Calculate the integral \begin{align} \int_{0}^{1} \frac{ \ln(\sqrt{x} - \sqrt{1-x}) }{ \sqrt{x} } \ dx \end{align} and show the value is negative.
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3answers
56 views

Determining $\lim_{n\to\infty}\left(n^{\tfrac{1}{n}}-1\right)^n$ with only elementary math

I am trying to find this limit: $$\lim_{n\to\infty}\left(n^{\tfrac{1}{n}}-1\right)^n,$$ I tried using exponential function, but I see no way at the moment. I am not allowed to use any kind of ...
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2answers
83 views

$\ln(n)/n<1/2$ proof without calculus or any kind of advanced mathematics

Is it possible to show that $\ln(n)/n<1/2$, for all natural numbers $n$ without using calculus, but just some elementary math? Induction is allowed. I was trying to show equivalently that ...
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0answers
45 views

How do I apply law of log here

My function is defined as How do I find log M(t)?
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0answers
25 views

applying logarithm law question

Here is my equation (below) on which I am applying log $X=\frac{a}{b}\left ( c-d \right )$ so far I applied it as $\log X=\log(a)-\log(b)+\left [ \log\left ( c \right )-\log\left ( d \right ) ...
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1answer
25 views

If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$. [duplicate]

If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$. I tried to separate the terms first and I got $\dfrac12 (\log(1+\log x) - \log(1-\log x))$. The answer is $\dfrac1{x(1-\log x)^2}$.
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1answer
26 views

An efficient technique to test if an exponential of logs gives an integer

Is there an efficient way of testing if the resulting value of an exponential gives an integer without actually expanding the equation. For example: $ {\log(12) - \log(4)}=1.09861\ldots $ and is a ...
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0answers
25 views

does $\ln{a}/\ln{b}=\log_ba$ still stand when $a,b\in\Bbb C$? ($b\ne1$)

I've heard that the property of logarithm becomes to have some differences with complex numbers. I'm not sure whether I should apply or not the property that's used with positive numbers.
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0answers
62 views

Logarithm and “basic” functions.

To express the antiderivatives of $\frac{1}{x}$, we cannot apply the formula $\int x^n dx=\frac{x^{n+1}}{n+1}+C$ and we need to introduce a new function, the logarithm. But how can we prove that ...
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1answer
37 views

Sketching the graph of $y =\ln(4-x)$

$y = \ln(4 - x) $ This graph has two operations applied to the $\ln x$ graph - a reflection and a translation. If you reflect the graph in the $y$-axis first, and then shift the graph 4 units to ...
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1answer
23 views

Function that represents growth with specific specs

I'm looking for a function that represents growth based on the following specs: In a range of 365 days the function may grow from (almost) zero to a maximum of 1. It should have some kind of ...
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1answer
18 views

Applications of logarithmic functions in shapes and geometries

As I understand this logarithmic functions are a family of functions where the equation for $f(x)$ is written like so $$\begin{align} f(x) = & \log_{a} x \\ & \mathtt{where\ }a\mathtt{\ is\ ...
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3answers
381 views

Solving exponential equations using logarithms

This is the equation that I am having troubles with: $$\large x^{\large\log_{10}5}+5^{\large\log_{10}x}=50$$ So the first thing I do, I logarithm the whole expression with $\log_{10}$. So I get: ...
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1answer
85 views

Logarithm of $\frac{a^k}{a^k-1}$

On a question on this site there is an explanation of the algorithm Knuth gives in The Art Of Computer Programming to compute an approximation of $y = \log_bx$. Now, I understand why it works; ...
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4answers
83 views

Arctan equalling a logarithm?

Is $$\int \frac{dx}{ax^2+bx+c}=\frac{1}{a}\int{dx}{(x-\alpha)(x-\beta)}=\frac{1}{a} \int \left( \frac{ \frac{1}{\alpha-\beta} }{x-\alpha} - \frac{\frac{1}{\alpha-\beta}}{x-\beta}\right)=\frac{1}{a} ...
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3answers
101 views

Logarithm Fraction Contest Math Question

The question is as follows: If $\dfrac{\log_ba}{\log_ca}=\dfrac{19}{99}$ then $\dfrac{b}{c}=c^k$. Compute $k$.
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1answer
82 views

Tighter logarithmic inequality

There is a well-known lower bound for $$ x\log{1+x\over x}\geq {x\over1+x} $$ for $x\geq0$. I know a tighter lower bound on the same domain $$ x\log{1+x\over x}\geq{2x\over1+2x}\geq {x\over1+x}. $$ It ...
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1answer
45 views

trouble with infinite values from exp() and log()

I'm writing a function for Gaussian mixture models with spherical covariance structures--ie $\Sigma_k = \sigma_k^2 I$. This particular function is similar to the ...
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0answers
84 views

What did “logarithm” initially mean? [duplicate]

I just read that logarithms were not initially defined in terms of their inverse relationship to exponential functions (and that Euler was the first to develop them in this way). So how were they ...
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1answer
72 views

How to simplify $3^{(2\log_335)}$

$3^{2\log_35}$ How do I simplify this? This is what I have done so far: $2\log_35=\log_35^2=\log_3(25)$ $3^{\log_3(25)}$ What do I do from here? And the answer is one of these mixed solutions: ...
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2answers
103 views

Non-integral power of a singular matrix

I know, that if $A$ is nonsingular matrix, so $\det{A} \ne 0$, then $A^p=\exp\left(p\ln A\right)$ is true for any real exponent, but what about if $A$ is singular? Then $A$ has a zero eigenvalue, so ...
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1answer
46 views

Logarithmic Contest Question

The Problem was as follows: Define $\log*(n)$ to be the smallest number of times the log function must be iteratively applied to $n$ to get a result less than or equal to $1$. For example ...
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3answers
151 views

Something about $\frac{\log x}{x}$

Denote $\log x = \log_ex$. Let's consider the below function $$\frac{\log x}{x}$$. Apparently, It's maximum is $\frac{1}{e}$. and strictly increasing in $(0,e]$, strictly decreasing in $[e,+\infty)$. ...
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0answers
39 views

Are generating functions ever analytic for logarithmic series?

Given a series $s_n = \ln(n) f(n)$ where $f(\cdot)$ is an elementary analytic function which does not involve the logarithm. More precisely $f$ can have simple poles but no branch cuts or essential ...
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1answer
30 views

Exponential percentage decrease based on time

I have a bar that shows the time left for a task to finish and I want it to decrease faster as it gets closer to the end time. Example: Let's assume that the total time required for Task A to ...
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1answer
78 views

$\iiint \frac{1}{x^2+y^2+(z-2)^2}dA$ where $A=\{x^2+y^2+z^2 \leq 1\}$ check my answer!

I would like someone to review my solution please, the original question is to calculate $\iiint \frac{1}{x^2+y^2+(z-2)^2}dA$ where $A=\{x^2+y^2+z^2 \leq 1\}$ What I did: First I changed variables ...
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1answer
84 views

Log Log Integrals II

The integral \begin{align} I_{4} = \int_{0}^{1} \ln(1-x) \ \ln^{2}\left( \ln\left(\frac{1}{x}\right) \right) \ \frac{dx}{x} \end{align} can be expressed as \begin{align} I_{4} = \zeta^{''}(2) - ...
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2answers
35 views

Given that $\log_2(x)=p$ and $\log_4(y)=q$, how do I evaluate $\log_x(4y)$?

Given that $\log_2(x)=p$ and $\log_4(y)=q$, how do I evaluate $\log_x(4y)$? There were some other questions like this and I applied this formula to them $\log_a(xy) = \log_a(x)+\log_a(y)$. However, in ...
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1answer
34 views

Convergence $\sum_{n=2}^{\infty} { \frac{\sqrt[n]{n^{p}}}{n\ln{n}} }$

Help please. I need to check the convergence $$\sum_{n=2}^{\infty} { \frac{\sqrt[n]{n^{p}}}{n\ln{n}} }$$Tried with Leibniz, but can't check monotony.
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3answers
109 views

Summation with Ceilinged Logarithmic Function

According to Johann Blieberger's paper - "Discrete Loops and Worst Case Performance" (1994): $$ \sum_{i = 1}^{n}\left \lceil \log_2{(i)} \right \rceil = n\left \lceil \log_2{(n)} \right \rceil - ...
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1answer
35 views

Asymptotic behaviour of a couple of special functions (features exponentials and logarithms)

I'm dealing with a couple of functions: $n \log n$, $( \log \log n)^{ \log n}$, $( \log n)^{ \log \log n}$, $n e^{\sqrt{n}}$, $( \log n)^{ \log n}$, $n 2^{ \log \log n}$, $n^{1+1/( \log \log ...
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1answer
68 views

How to simplify $\ln^2\left(x\right)+2 \ln x-3$

I dont know how to simplify $\ln^2\left(x\right)+2 \ln x-3$ I dont know how to get $(\ln(x)+1)(\ln(x)+3)$ But I am stuck and don't really know how to do that. I tried something like this: $2\ln ...
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4answers
79 views

Solve for $x$ in $2\log(x+11)=(\frac{1}{2})^x$

Solve for $x$. $$2\log(x+11)=(1/2)^x$$ My attempt: $$\log(x+11)=\dfrac{1}{(2^x)(2)}$$ $$10^{1/(2^x)(2)}= x+11$$ $$x=10^{1/(2^x)(2)}-11$$ I'm not sure what to do next, because i have one $x$ in ...
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1answer
76 views

Upperbound for $ \sum _{i=1}^N a_i\ln a_i $

It's easy to prove that following upperbound is true: $\sum_{i=1}^N a_i\ln a_i \le A \ln A$, where $\sum_{i=1}^N a_i=A$ and $ a_i\ge 1$ I'm wondering, is there stronger upperbound?
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1answer
94 views

Unable to comprehend a connection between two equations

I was reading this paper and got stuck at the transition from Equation (13) to Equation (14) (p. 16/17). We got a function of the form: $y(t)=k(t)^{\alpha}h(t)^{\beta}$ We know it grows from zero ...
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1answer
39 views

Log-space probability of a log-space probability not occurring

Normally the probability of some probability $p$ not occurring would be $1-p$. However, I'm working with very small probabilities and therefore must work with $p$ in $\log$ space (Ie. I'm working with ...
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1answer
48 views

Could somebody validate my proof regarding the limit of $\ln(x_n)$ when, $x_n$ tends to $a$?

So, let me cearly state the problem: Let $(x_n)$ be a convergent sequence, with: $ x_n > 0 $, $\forall n$, n natural number, and $x_n \to a$, with $a>0$. Then $\ln{x_n} \to \ln{a}$. Here is my ...
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1answer
37 views

Solving $Ae^x=Bx$ analytically, where $A$ and $B$ are constants?

This equation mixes both exponential terms and linear terms, something which I do not know how to deal with. Any pointers?
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1answer
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Struggling with a Form of Logarithm question during my revision

I am doing AS Mathematics In the UK under the examining board edexcel. I came across this question in a List of exam questions given to me by my teacher However I can't work out how to do it. ...
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1answer
60 views

How did Napier rounded his logarithms?

How did Napier round his logarithms? Wikipedia says: By repeated subtractions Napier calculated $(1 − 10^{−7})^L$ for $L$ ranging from 1 to 100. The result for $L=100$ is approximately $0.99999 = ...
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2answers
191 views

Properties of Logarithms

How do you simplify the following expression? $$\log\left(3^{(5^7)}\right)$$ I know that logarithms are like the inverse of exponents, but are there any tricks to simplify powers inside logarithms? ...
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2answers
79 views

Finding unknown in log equation

I was given a log equation: $$D = 10 \log (I/I_0) $$ $I$ is the unknown in this case, $I_0 = 10^{-12}$ and $D = 89.3$. I did the following steps: $$ \begin{aligned} \ 89.3 &= 10 \log ...
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4answers
171 views

What did Johann Bernoulli wrong in his proof of $\ln z=\ln (-z)$?

Some people say, Johann Bernoulli has proven $\ln z=\ln (-z)$ in the following way $$\ln ((-z)^2 )=\ln(z^2)\;\;\;\Rightarrow\;\;\;2\ln(-z)=2\ln z\;\;\;\Rightarrow\;\;\;\ln (-z)=\ln z$$ While the ...
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1answer
36 views

How to rewrite this logarithmic update rule

I tried to rewrite the equation given below. I get stuck getting rid of the $ P(n|z_{1:t})$ on the left side. How can this be done? $$ P(n|z_{1:t}) = \left[1+ \frac{1-P(n|z_{t})}{P(n|z_t)} ...
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1answer
97 views

Equal spaced points in a logarithmic graph

I am plotting a graph with the x-axis as logarithmic. I want to select 10 point that are equally spaced in a logarithm scale. How can I determine the values if we have the range from 100 to 10000?
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29 views

Logarithmic Properties for Equation [closed]

What are the steps/logarithmic properties used to solve the equation 2^n=n^8? Thanks!
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1answer
43 views

How is $\arctan(\sinh(x * \pi))$ the inverse of $\log(\tan(x)) / \pi$

How is $$\arctan(\sinh(x * \pi))$$ the inverse of $$\frac{\log(\tan(x))}{\pi}$$ What is the relationship between $\log(x)$ and $\sinh(x)$. I guess is what my real question is.
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3answers
56 views

Why $\ln 2=\ln 1.075^t\implies \ln 2=t\ln 1.075$

Why $$\ln 2=\ln 1.075^t\implies \ln 2=t\ln 1.075$$
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2answers
74 views

Solve numerical system of nonlinear equations?

I need to solve a nonlinear system of equations that looks like this ...