Questions related to real and complex logarithms.

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0
votes
1answer
108 views

Finding the transfinite diameter of the level sets of complex logarithm

Given a simply-connected domain $|g(z)|\ge C$ how can I find the analytic conformal mapping guaranteed by the Riemann mapping theorem? In particular I'm interested in finding the transfinite diameter ...
0
votes
1answer
15 views

What is the logaritmic form of $v=Ae^{Bi}$

I am reading a scientific paper, which uses a model of the form $v=Ae^{Bi}$ and then it says that this model has the following logarithmic form $\ln (v) = Bi + ln(A)$ where A is a constant. But the ...
0
votes
2answers
37 views

How to use the Comparison Test to investigate the convergence of $\sum (\ln n)/n^\alpha$?

Let $$\sum\limits_{n=1}^\infty \frac{\ln n}{n^\alpha}, \alpha\in\Bbb{R}$$ I need to investigate the convergence of this series. I've read that since the series is positive for all $n$ then it ...
0
votes
1answer
12 views

Multivariable-calculus, logarithms

I got the function $f(x,y)=\ln(1+x^2+y^2)$. There are three tasks to answer. a)Decide the function´s stationary points and classify them if possible. Here I got the answer to $(0,0)$ is a local ...
7
votes
1answer
96 views

How to solve $\log _{x^{2}-3}(x^{2}+6x)<\log _{x}(x+2)$?

How to solve the following inequality $$\log _{x^{2}-3}\left(x^{2}+6x\right)<\log _{x}(x+2)\ ?$$
0
votes
2answers
51 views

A Method For Calculating Large Exponents Quickly

I've derived a formula for calculating large exponents quickly: $$a^b = 2 \cosh( - b \log( a ) )$$ My question is: Has anyone seen anything similar? I am curious if either it's novel OR if I have ...
3
votes
5answers
202 views

Can this log question be simplified?

$ { 2^{log_3 5}} -  {5^{log_3 2}}.$ I don't know any formula that can apply to it or is there a formula?? Even a hint will be helpful.
1
vote
1answer
44 views

Find real-valued sequences $x(n)$ for which $c^{x(n)} = o(1/n )$

For which $x=x(n)$ does it hold that $$c^x = o\left(\frac{1}{n}\right)$$ where $c\in(0,1)$ is a constant. So clearly, for $x=n$, this is true. But for which $x =o(n)$ does this hold? I thought ...
0
votes
1answer
27 views

How to scale a equation e.g. by log

I'm currently trying to scale an equation since the numbers I have to calculate with are pretty large and Matlab outputs Infinity (Inf). However, the question here is more about the mathematics behind ...
4
votes
1answer
27 views

Generalize logarithmic coincidences

After playing around with logarithms, I've found the following coincidences: $\log_{10}{2} \approx 0.3$, since $2^{10} \approx 10^3$, and $\log_{10}{5} \approx 0.7$, since $5^{1000} \approx ...
2
votes
4answers
67 views

How to solve this kind of equation $(x^y=y^x)$

I'm little bit stuck with this system of equations : $x^y=y^x$ and $x^3=y^2$ An obvious solution is $(x,y) = (1,1)$ but what about the solution $(9/4,27/8)$ ? I know the relation $a^r=e^{r ...
1
vote
3answers
57 views

How to use the logarithm method to solve $18^{4x-3}=(54\sqrt{2})^{3x-4}$ for $x$?

What value will satisfy this equation: $$18^{4x-3}=(54\sqrt{2})^{3x-4}$$ Please use the logarithm method. I am having a problem in expressing $54\sqrt{2}$ in the power of $18$. My book simply ...
3
votes
4answers
75 views

Exponential equation: $2e^{-x} - e^{-2x}=0.$ [closed]

$2e^{-x} - e^{-2x}=0.$ the correct answer is $x=-\ln2$. How do I get there?
0
votes
0answers
15 views

Normalizing Data for Graph

Firstly, sorry for the long post, but I must be detailed in my explanation here. This is a computer science heavy topic, and I've posted it on the CS section of Stack Overflow already, but the main ...
5
votes
3answers
118 views

Solve for $x$ in the equation [closed]

Please help me to solve for x using maybe logarithm or exponential rules (or both) $$ 5^x=2 \cdot 3^x $$
1
vote
4answers
116 views

Solving the logarithimic inequality $\log_2\frac{x}{2} + \frac{\log_2x^2}{\log_2\frac{2}{x} } \leq 1$

I tried solving the logarithmic inequality: $$\log_2\frac{x}{2} + \frac{\log_2x^2}{\log_2\frac{2}{x} } \leq 1$$ several times but keeping getting wrong answers.
0
votes
2answers
43 views

Dealing with Logarithms. $\log(b^x + a) = \log(c)$

What methods/techniques are available to solve for x in the following type of situation: $$ \log(b ^x+a)=\log( c ) $$ The only log methods I have been exposed to are using the power laws and bring x ...
1
vote
0answers
20 views

Deriving cost function using MLE :Why use log function?

I am learning machine learning from Andrew Ng's open-class notes and coursera.org. I am trying to understand how the cost function for the logistic regression is derived. I will start with the cost ...
3
votes
2answers
70 views

Solution for this Logarithmic Equation

Recently I was going through a problem from the book Problems in Mathematics - *V Govorov & P Dybov* . $$(x-2)^{\log^2(x-2)+\log(x-2)^5-12}=10^2\log(x-2)$$ I tried solving by first considering ...
1
vote
5answers
78 views

What is the reason to introduce and study logarithmic functions?

I don't understand why logarithms exist when we have exponential functions. Exponential functions seem to be an easier and less convoluted way to write something. Why invent logarithms to do something ...
0
votes
0answers
45 views

Is this double limit for logarithms true?

Mathematica knows that: $$\gamma = \lim_{n\to \infty } \, \lim_{s\to 0} \, \left(\int \frac{(s+1)^{-\exp (n)-1}+s-1}{s} \, ds+\frac{(s+1)^{-n-1}+s-1}{s}\right)$$ Where $\gamma$ is Euler Gamma ...
2
votes
1answer
80 views

How does the Sin and Cos scale on a slide rule work and what is the formula for it?

As I described in this question, I am trying to make a printable slide rule (similar to the slide rule provided in this SciAm article). I have made most of the parts of it but I can't make the Sin and ...
1
vote
2answers
131 views

$\int_0^1\frac{1-t}{(t-2)\ln t}\,dt$ integral

I have two related questions. The first is: Is there a closed form expression for: $$\int_0^1\frac{1-t}{(t-2)\ln t}\,dt\approx0.507834$$ I know that there are some very superb integrators on this ...
1
vote
0answers
58 views

Approximation for the convolution of normal and lognormal distributions

$$X \sim \ln\mathcal{N}(\mu_X,\,\sigma_X)$$ $$Y \sim \mathcal{N}(0,\,1)$$ $$Z = X + Y$$ I want to find the probability density functions and cumulative distribution functions of $Z$. As the below is ...
0
votes
1answer
70 views

Solving the system with logarithms

I tried solving the system $ \begin{cases} (4x)^{\log_2 (2y)} = 64 \\ (8y)^{\log_2 (2y)} = 256 \end{cases} $ several times but still keep getting wrong solutions.
0
votes
0answers
55 views

Why $0^0$ is not equal to $1$? [duplicate]

Suppose that 0^0= x Then, taking logarithm in both side we got: 0*log 0= log x Left hand side is equal to 0, so: 0= log x Then, the satisfied value of x is only 1. So, 0^0= 1 But, in calculus ...
4
votes
3answers
44 views

Mechanic method to draw a logarithmic spiral?

I'm in the need to draw (more like to use a wood router to carve a groove) a logarithmic spiral in a piece of wood. So, I got a router that is attached to a stick, I draw a circle by rotating the ...
3
votes
4answers
101 views

Solving $e^{4x}+3e^{2x}-28=0$

How to solve this equation: $$e^{4x}+3e^{2x}-28=0$$ I don't know how to solve this problem. I read over another example, $e^{2x}-2e^x-8=0,$ and it said that $e^{2x}$ is $e$ to the $x$ squared, ...
15
votes
0answers
217 views

Ramanujan log-trigonometric integrals

I discovered the following conjectured identity numerically while studying a family of related integrals. Let's set $$ R^{+}:= \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x} ...
0
votes
0answers
17 views

Linearizing non-linear least squares: Problem with derivatives

We want to approximate $$y_i \approx a b^{x_i}$$ and thus have $$S=\sum_{i=1}^m (ab^{x_i}-y_i)^2$$ as least squares error term. This term is not linear in b, so it is not easy to calculate its ...
1
vote
2answers
61 views

Having trouble solving $\log (x − 21) = 2 − \log x$ for $x$

I'm having trouble with this problem: $\log (x − 21) = 2 − \log x$, solve for $x$. I'm coming up with $x=-5$ but that can't be right.
0
votes
2answers
39 views

Limit of log with factor

Problem If $$\lim_{n \rightarrow \infty} f(n) - \log n - \log \log n = y$$ where $f$ is some function, does this imply that $$ \lim_{n \rightarrow \infty} f(n) - \log (xn) - \log \log n = y$$ for ...
1
vote
2answers
67 views

Neither $\log x$ nor $\exp(x)$ are rational functions [closed]

(a) Prove that $\log x$ cannot be expressed in the form $f(x)/g(x)$ where $f(x)$ and $g(x)$ are polynomials with real coefficients. (b) Prove that $e^x$ cannot be expressed in the form $f(x)/g(x)$ ...
3
votes
3answers
60 views

Evaluate the limit of $\ln(\cos 2x)/\ln (\cos 3x)$ as $x\to 0$

Evaluate Limits $$\lim_{x\to 0}\frac{\ln(\cos(2x))}{\ln(\cos(3x))}$$ Method 1 :Using L'Hopital's Rule to Evaluate Limits (indicated by $\stackrel{LHR}{=}$. LHR stands for L'Hôpital Rule) ...
3
votes
1answer
26 views

Why does $\lim\limits_{N \rightarrow \infty}{\sum_{i=1}^{N}\frac{1}{\frac{N}{1-\epsilon}-i}}$ converge to $\log\left[\frac{1}{\epsilon}\right]$?

while playing around with my equations, i found that the following has to hold for my universe to be consistent: $$\lim_{N \rightarrow ...
2
votes
3answers
95 views

Evaluate $\int_{1}^{\infty} \frac{\ln{(2x-1)}}{x^2} $

$$\int_{1}^{\infty} \frac{\ln{(2x-1)}}{x^2} dx$$ My approach is to calc $$\int_{1}^{X} \frac{\ln{(2x-1)}}{x^2} dx$$ and then take the limit for the answer when $X \rightarrow \infty$ However, I must ...
0
votes
0answers
29 views

Integrating the logarithm of a function including a square root of a second degree polynomial

I have been trying for some time to calculate the following integral: $$\int \ln\left(k+\sqrt{ax^2+bx+c}\right)\ dx$$ where k, a, b and c are real numbers. I have tried several strategies, but without ...
1
vote
3answers
26 views

Common logarithm question

I'm studying logarithms and am doing an exercise where you're supposed to evaluate the solutions of common logarithms without using a calculator. I'm very stuck on this one particular question. I know ...
1
vote
3answers
30 views

How many solutions to quadratic logarithms?

For a given Log equation with a quadratic $x$, such as $$F(x)=\log(x^2)$$there appear to be two $x$ values, for every $F(x)$, a positve and a negative. However, if $F(x)$ is ...
1
vote
0answers
26 views

Can difference of the $log$ function be approximated?

I am currently trying to optimize a problem. $$\text{ArgMax}_x \log(1+f_1(x))-\log(1+f_2(x))$$ Due to the fact that $\log (x)$ is a monotonic increasing function, this is equivalent as to ...
8
votes
2answers
169 views

A closed form for $\int_{0}^{\pi/2}\frac{\ln\cos x}{x}\mathrm{d}x$?

The following integrals are classic, initiated by L. Euler. \begin{align} \displaystyle \int_{0}^{\pi/2} x^3 \ln\cos x\:\mathrm{d}x & = -\frac{\pi^4}{64} \ln 2-\frac{3\pi^2}{16} ...
1
vote
2answers
39 views

Check my proof: Big O notation

I was asked the following: We are given the functions $f(n)=n^{10\log(n)}$ and $g(n)=(\log (n))^n$. Which of the following statements is true: $f(n)\in\mathcal{O}(g(n))$, $f(n) \in ...
0
votes
1answer
20 views

Finding the leading exponent of a binary number

Let's say that the binary representation of a number $k$ is $2^{X_n} + 2^{X_{n-1}} + \dots + 2^{X_0}$ with each term in this polynomial having a $1$ or $0$ multiplied to it (I just haven't showed them ...
0
votes
2answers
62 views

Does the definition range remains the same?

In solving this inequality (transcribed from here) $$\left(\frac23\right)^{\log_{0.5}(x^2+4x+4)}<\left(\frac94\right)^{\log_2(x^2-3x-10)}$$ we eventually reach the point where $ ...
2
votes
3answers
38 views

inverse of quadratic log functions

Can a Log function with a quadratic have an inverse function? The specific question is to find the inverse of $$f(x) = \log_2(x^2-3x-4)$$ The function already fails the horizontal line test, but ...
8
votes
1answer
105 views

Feynman's Algorithm for computing a logarithm of a number in [1,2]

I came upon the following quote concerning Feynman (the entire essay this is from can be found here): Consider the problem of finding the logarithm of a fractional number between 1.0 and 2.0 (the ...
11
votes
2answers
364 views

Evaluating $\int_0^\pi\arctan\left(\frac{\ln\sin x}{x}\right)\mathrm{d}x$

I found the following integral as a by product of another one. It has a nice closed form. $$ \int_{0}^{\pi} \arctan\left(\ln\left(\sin x \right) \over x\right)\,{\rm d}x $$ Mathematica and ...
1
vote
1answer
42 views

Logarithmic integral and natural numbers.

Prove these two relations: $$\text{li}(k+1)+k-\log (k)-\gamma = \int_0^k \left(\int_1^2 \frac{(s+1)^{n-1}+s-1}{s} \, dn\right) \, ds$$ $$n = \lim_{s\to 0} \, \frac{(s+1)^{n-1}+s-1}{s}$$ ...
0
votes
2answers
43 views

Logarithmic question

How is: $$n^{\Large\frac 1{\lg n}} = 2\ ? $$ I don't understand this is their any formula to calculate this what is the difference between $\lg n$ & $\log n$? Is logarithm base $2$ ?
2
votes
2answers
147 views

Solve for $x$: $\frac1e = e^{2x}$

I tried making it to $e^{-1} = e^{2x}$ and had the exponents equal each other $-1=2x$ and the I solved for $x$, making it $x=-1/2$, but that answer is wrong. please help I don't know why that ...