Questions related to real and complex logarithms.

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2
votes
4answers
91 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, ...
1
vote
1answer
21 views

Testing the convergence of the series $\sum 1/(k^q (\ln k)^p)$

Determine all values of $p$ and $q$ for which the following series converges: $$\sum_{k=2}^{\infty} \frac{1}{k^q (\ln k)^p}$$ Hints : Consider the three case $q>1$, $q=1$, $q<1$. I ...
0
votes
1answer
31 views

Determine the realtions ($\mathcal{O}$,$\Theta$,$\Omega$ ) between $f(n) = \ln(n^{c} + n^{d})$ and $g(n)=\ln(n^{a} + n^{b})$

I am trying to determine the realtions ($\mathcal{O} $,$\Theta$,$\Omega$ ) between : $$f(n) = \ln(n^{c} + n^{d})$$ $$g(n)=\ln(n^{a} + n^{b})$$ Note: $a,b,c,d>0$ I need some advice how to use the ...
1
vote
3answers
62 views

Estimating the natural logarithm from both sides: $1/(a+1)<\ln((1+a)/a) <1/a$

I must prove that for all $a>0$ $$\frac{1}{a+1}<\ln{\frac{1+a}{a}}<\frac{1}{a}$$ Can anyone help me?
3
votes
0answers
42 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
12 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
46 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 ...
1
vote
2answers
44 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
36 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
1answer
50 views

Solving $F(x) = 0.3$ where $F(x) = 1-\frac{200^{ 2.5 }}{ x^{2.5}}$

Consider the function $$F(x) = 1-\frac{200^{ 2.5 }}{ x^{2.5}}.$$ We want to solve $F(x) = 0.3$. Since $ F(x) = 0.3$ then we can say, $2.5 \ln (\frac{200}{x}) = 0.7$. $\ln(\frac{200}{x}) = 0 .28$. ...
-2
votes
1answer
44 views

Lower bound for $\ln x$ using Lagrange's mean value theorem or Rolle's theorem

I have to prove this inequality. $$ \ln x>\frac{2(x-1)}{x+1} \hspace{15pt}, \hspace{15pt}\text{where}\hspace{5pt}x>1 $$ using either Lagrange's mean value theorem or Rolle's theorem. Can ...
-1
votes
2answers
2k views

Writing an equation for a log function given the graph

I have the following graph for a logarithmic function $f$: I don't know any thing about writing an equation for a logarithmic function by knowing it's graph. All what I know is how to draw a graph ...
3
votes
3answers
55 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) ...
1
vote
2answers
57 views

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

(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
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
84 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 ...
3
votes
1answer
726 views

How to prove if log is rational/irrational

I'm an English major, now doubling in computer science. The first course I'm taking is Discrete Mathematics for Computer Science, using the MIT 6.042 textbook. Within the first chapter of the book's ...
3
votes
1answer
478 views

What is the opposite of logarithmic scale?

If one wants to plot a variable whose value changes drastically, it makes sense to plot its order of magnitude. That way, the smaller values in the plot aren't squished all the way at the bottom. This ...
4
votes
3answers
78 views

Solving $7^{2x}\cdot4^{x-2}=11^x$

I am trying to solve this equation but I am stuck little bit right now. This is how I did it: $$ 7^{2x}\cdot4^{x-2}=11^x\\ \text{log both sides}\\ \log(7^{2x}\cdot4^{x-2})=\log(11^x)\\ \log(7^{2x}) + ...
0
votes
1answer
31 views

Finding the exponent of $2$ such that $x \cdot 2^a$ is as close to $1$ as possible

How do I find an exponent of $2$ that when multiplied with another number would bring the result closest to the positive side $1$? Like this: $y = x \cdot 2^a$, where $y\ge 1$ has to be as small as ...
0
votes
1answer
37 views

Characteristic function of logarithm of random variable

If I know the characteristic function $\phi_X(t)$ of a random variable $X>0$, how can I write the characteristic function $\phi_Y(t)$ of $Y=\log(X)$? I know that $\phi_X(t)=E[e^{itX}]$ and ...
10
votes
2answers
319 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
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 ...
0
votes
0answers
26 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
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
25 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
152 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} ...
0
votes
2answers
58 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 $ ...
8
votes
1answer
101 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 ...
1
vote
2answers
38 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
1answer
69 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: ...
7
votes
1answer
194 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 - ...
2
votes
3answers
36 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 ...
2
votes
1answer
128 views

What is ${\mathfrak{R}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$?

This is a new integral that I propose to evaluate in closed form: $$ {\mathfrak{R}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$$ where $\log (z)$ denotes the principal value of ...
5
votes
0answers
160 views
+50

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 $$\sum_{n=2}^{\infty} \frac{1}{\ln^n{n}} \approx 3.2426094109 $$ ...
4
votes
1answer
91 views

Prove this polylogarithmic integral has the stated closed form value

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

Logarithmic integral and natural numbers.

Prove these two relations: $$\text{li}(k+1)+k-\log (k)-\gamma = \int_0^k \left(\int_1^2 \frac{(s+1)^{n-1}+s-1}{s} \, dn\right) \, ds$$ $$n = \lim_{s\to 0} \, \frac{(s+1)^{n-1}+s-1}{s}$$ ...
6
votes
2answers
278 views

A series with only rational terms for $\ln \ln 2$

We all know that $$ \ln 2 = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}. $$ Do you know a series with only rational terms for $$\ln \ln 2 = ?$$ Let's exclude base expansions with non ...
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
137 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 ...
14
votes
4answers
563 views

A closed form for $\int_{0}^{\pi/2} x^3 \ln^3(2 \cos x)\:\mathrm{d}x$

We already know that \begin{align} \displaystyle & \int_{0}^{\pi/2} x \ln(2 \cos x)\:\mathrm{d}x = -\frac{7}{16} \zeta(3), \\\\ & \int_{0}^{\pi/2} x^2 \ln^2(2 \cos x)\:\mathrm{d}x = ...
0
votes
1answer
42 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 ...
1
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, ...
5
votes
2answers
90 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 ...
0
votes
0answers
26 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
77 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 ...
2
votes
2answers
55 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 ...
3
votes
3answers
104 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 ...
8
votes
2answers
218 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 ...