Questions related to real and complex logarithms.

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2answers
45 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
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0answers
23 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
72 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
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5answers
80 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
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0answers
47 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
85 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
137 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
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0answers
72 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
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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.
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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
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3answers
52 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
105 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, ...
33
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1answer
381 views
+100

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
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0answers
18 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
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2answers
65 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
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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
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2answers
68 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
61 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
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1answer
27 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
100 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
30 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 ...
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3answers
30 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
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3answers
31 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
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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
201 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
46 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
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2answers
63 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
39 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
110 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 ...
13
votes
2answers
414 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
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1answer
45 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
148 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 ...
0
votes
1answer
43 views

Why, or why not, is $5^{log_3(n)} \in \mathcal{O}(n^2)$?

Why, or why not, is $5^{\log_3(n)} \in \mathcal{O}(n^2)$ ? I tried transforming the logarithm to a base of 5, so that the logarithm and power cancel each other out. However, when I try to so I get ...
7
votes
1answer
234 views

An equivalent for $\int_0^1\left(\frac{1}{\log x}+\frac{1}{1-x}\right)^n\;dx$

Set $$ I_n :=\int_0^1\left(\frac{1}{\log x} + \frac{1}{1-x}\right)^n \:\mathrm{d}x \qquad n=1,2,3,.... $$ We have $$I_1 =\gamma, \quad I_2 =\log (2 \pi) - \frac 32, \quad I_3 = 6 \log A - ...
0
votes
0answers
27 views

Can not determine scale of x-axis of attached graph

I am trying to digitize the data from the graph in this link. I thought the $x$-axis was in $\log_{10}$ scale but after trying to digitize this way the points seemed off. I also tried digitizing in ...
1
vote
1answer
81 views

Why would $\forall x\log(x) = 0 \implies 2^\frac{1}{n} - 1 \leq \frac{\epsilon}{n}$ for large $n$

Why would $\forall x\log(x) = 0 \implies 2^\frac{1}{n} - 1 \leq \frac{\epsilon}{n}$ for large $n$? I'm reading a calculus text which used this in a reductio to prove the log function is nontrivial and ...
5
votes
2answers
105 views

How is $ \sum_{n=1}^{\infty}\left(\psi(\alpha n)-\log(\alpha n)+\frac{1}{2\alpha n}\right)$ when $\alpha$ is great?

Let $\psi := \Gamma'/\Gamma$ denote the digamma function. Could you find, as $\alpha$ tends to $+\infty$, an equivalent term for the following series? $$ \sum_{n=1}^{\infty} \left( \psi (\alpha ...
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vote
4answers
1k views

What's wrong with my aproach to solving this equation with multiple logarithms?

A question I was faced with asked "For which $x$ is $\log_{10}(x)^{\log_{10}(\log_{10}(x))}= 10,000$?" My instincts tell me I can say $$\log_{10}(x)=10$$ and $$\log_{10}(\log_{10}(x))=4$$ However, ...
2
votes
2answers
59 views

Integrating 1/x

The standard definition of integrating $\frac{1}{x}$ is: $$ \int \frac{dx}{ax + b} = \frac {1}{a} \ln |ax + b| + K $$ Now, if I'm understanding the "constant factor rule", that is: $$ \int k ...
1
vote
2answers
42 views

Proof that $\log(a^b) = b\log a$ when $\log$ is defined by an integral

When $\log a$ is defined as $\displaystyle\int_1^a\frac{dx}x$, then how does one prove that $\log(a^b)=b\log a$? I will post an answer here that is identical to an answer I posted to another ...
0
votes
4answers
85 views

Proving $\log(b^a) = a \log(b)$ using calculus

Sorry, this is a really simple question, but I'm trying to teach myself calculus and can't figure it out. If we define $\log(b) = \frac{db^x}{dx}(0)$ how does one prove $\log(b^a) = a\log(b)$? I ...
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vote
2answers
75 views

$-\varepsilon\log(x)\overset{?}{\geq} -\log(\varepsilon x)$

I'm refering to this proof: http://en.wikipedia.org/wiki/Quantum_relative_entropy#The_result In there it's stated that "Since the matrix $(P_{ij})_{ij}$ is a doubly stochastic matrix and $-\log$ is a ...
8
votes
2answers
244 views

A Binet-like integral $\int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{x^s }{1-x}\mathrm{d}x$

I met this integral $$ \int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{ \mathrm{d}x}{1-x} \qquad (*) $$ while evaluating this log-cosine integral. I made several ...
3
votes
3answers
127 views

$\int_{0}^{\pi/2}\ln\left(1+4\sin^4 x\right)\mathrm{d}x$ and the golden ratio

We already know that, for any real number $t$ such that $t\geq-1$, $$ \int_{0}^{\pi/2} \ln \left(1+t \sin^2 x\right) \mathrm{d}x = \pi \ln \left( \frac{1+\sqrt{1+t}}{2} \right). $$ Prove that ...
3
votes
1answer
47 views

$e$ and natural logarithms

How would you solve $6xe^{2x}+3e^{2x}=0$ for $x$ I tried: $\ln(e^{2x})=\ln(1/6x+3)$ $2x=\ln(1)-\ln(6x+3)$ $2x=-\ln(6x+3)$ but then I am stuck there. What am I missing?
6
votes
1answer
248 views

Closed form $\sum_{n=2}^{\infty} \frac{1}{\ln^n{n}}$ and $\sum_{n=2}^{\infty} \frac{n}{\ln^n{n}}$

Apologies if this has been asked before, but I was playing around with Wolfram Alpha and got approximations but not closed forms for $$\sum_{n=2}^{\infty} \frac{1}{\ln^n{n}} \approx 3.2426094109 $$ ...
5
votes
1answer
126 views

Prove this polylogarithmic integral has the stated closed form value

Question. Prove the following polylogarithmic integral has the stated value: $$I:=\int_{0}^{1}\frac{\operatorname{Li}_2{(1-x)}\log^2{(1-x)}}{x}\mathrm{d}x=-11\zeta{(5)}+6\zeta{(3)}\zeta{(2)}.$$ ...
2
votes
1answer
69 views

What's an intuitive way to compute summation of this series?

What's an intuitive way to compute $$\log(1)+\log (2)+\log (3)+\cdots+\log (n-1)+\log (n)$$ or for $n>a$ $$\log(a)+\log (a+1)+\log (a+2)+\cdots+\log (n-1)+\log (n) $$ I know the answer for ...