Questions related to real and complex logarithms.

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1
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3answers
68 views

Which is asymptotically larger $n^2 \log(n)$ or $n (\log(n))^{10}$?

Which is asymptotically larger $n^2 \log(n)$ or $n ( \log(n))^{10}$? I have tried by plugging in the values and $n^2 \log(n)$ turns out to be bigger. How can this be done analytically?
0
votes
5answers
56 views

What is the Inverse function of $y = 10^{-x}$? Steps are appreciated.

What is the inverse of $y = 10^{-x}$? These are my steps for the problem. Step 1 $y = 10^{-x}$. Step 2 $x = 10^{-y}$ by inverse substitution. Step 3 $10^y(x) = 1$. Step 4 $10^y = ...
8
votes
7answers
2k views

An alternative way to calculate $\log(x)$

How can I replace the $\log(x)$ function by simple math operators like $+,-,\div$, and $\times$? I am writing a computer code and I must use $\log(x)$ in it. However, the technology I am using does ...
0
votes
1answer
40 views

Can I take an exponent out of a sum?

For example, assuming we had a sum: $$\sum_{n=1}^m n^b \quad m,b\in\mathbb{N}$$ Is there any way to take the $b$ out of the sum? I tried taking the $\log_n$ of every value, add them together then ...
0
votes
3answers
40 views

Solve $\log_{1/4}{x}=\frac{3}{2}$

I want to solve $$\log_{1/4}{x}=\frac{3}{2}$$ Now I know the result is: $$\frac{1}{8}$$ but I am not sure how to get it. Any help would be greatly appreciated.
2
votes
4answers
50 views

Solve $\log_{1/3}(x^2-3x+3)≥0$

I want to solve $$\log_{1/3}(x^2-3x+3)≥0$$ Now I know the result is: $x ∈ <1;2>$, but i am not sure how to get it. My thoughts: $\frac{1}{3}$ to the power of positive number $= (x^2-3x+3)$, now ...
4
votes
1answer
125 views

A $\log \Gamma $ identity: Where does it come from?

$$\log \Gamma (n)=n\log n -n +\frac{1}{2} \log \frac{2\pi}{n}+\int_0^\infty \frac{2\arctan (\frac{x}{n})}{e^{2\pi x}-1} \,\mathrm{d}x$$ Is an identity that is derived from using Sterling's ...
3
votes
2answers
88 views

What is the integral of log(z) over the unit circle?

I tried three times, but in the end I am concluding that it equals infinity, after parametrizing, making a substitution and integrating directly (since the Residue Theorem is not applicable, because ...
0
votes
1answer
27 views

Solving logarithmic equation, different bases

What number do I need to multiply both sides with? I have worked for an hour on this but it is the first time I am using this website so it is impossible for me to write what I have already done. If ...
2
votes
3answers
2k views

Is there an approximation to the natural log function at large values?

At small values close to $x=1$, you can use taylor expansion for $\ln x$: $$ \ln x = (x-1) - \frac{1}{2}(x-1)^2 + ....$$ Is there any valid expansion or approximation for large values (or at ...
0
votes
2answers
119 views

Find the maximum possible value of $8(27)^{\log_{6}x}+27(8)^{\log_{6}x}-x^3$,where $x>0$

Find the maximum possible value of $8(27)^{\log_{6}x}+27(8)^{\log_{6}x}-x^3$,where $x>0$ Let $P(x)=8(27)^{\log_{6}x}+27(8)^{\log_{6}x}-x^3$ By using $AM-GM$ inequality on the first two terms ...
0
votes
2answers
26 views

Find number of digits of a number in another base

How can I solve this question: Given that $\log 3$ is about $0.48$, approximately how many digits are in the number $10^{150}$ if it were written in base $3$. Thanks!
1
vote
1answer
50 views

Finding Stationary Points of Natural Log Function

$$ f(x) = x - 2\ln(x^2 + 3) $$ I started by using the chain rule on $x^2 + 3$ which gives me $\frac{2x}{x^2} + 3$. At this point I tried to multiple $\frac{2x}{x^2} + 3$ by $x - 2$ - is this ...
2
votes
2answers
47 views

Rejecting a solution.

Why does it for $x^2=9$ we get two solutions, while if we use the "log both sides" property the negative solution is rejected? which method is true and why?
0
votes
0answers
48 views

integral involving error function (erf)

Does anybody know if a closed form of this integral exist? $\int \mbox{erf}(x) \ln(\mbox{erf}(x)) \Bbb dx$ where erf is so called error function. In case there is no closed form solution. Is it ...
1
vote
1answer
31 views

Logarithmic equation with variable both “free” and in logarithm

I am trying to calculate an area bordered by two functions and in the process I need to solve this equation: $$e^{-10x}=-2x+1$$ I make it into a non-exponential form: $$-10x=ln(-2x+1)$$ And now I am ...
1
vote
0answers
25 views

Derivation for Napier Logarithm

Something new that I came across called the Napier Logarithm. I was reading the book “Computing: A Historical and Technical Perspective”, and the book says that $$Nap.log x = 10^7(\log_e(10^7/x))$$ I ...
0
votes
3answers
76 views

Solving equations having both log and exponential forms

How can one Solve equations having both log and exponential forms: For eg... $e^x$ $=$ $\log_{0.001}(x)$ gives $x=0.000993$ (according to wolfram-alpha ...
1
vote
3answers
80 views

Better to memorize logarithm rules? And how?

Do people good at math totally memorize these logarithm rules below? If so, are there good mnemonics for this? I'm bad at math and I only memorize these rules really vaguely by rote, thus when needed, ...
1
vote
4answers
65 views

Simultaneous Equations (Stuck on the algebra)

Question: Solve the following simultaneous equations for real values of x and y $$ \left\{ \begin{array}{l} 9^{2x+y} - 9^x \times 3^y = 6 \\ \log_{x+1}(y+3) + \log_{x+1}(y+x+4) = 3 ...
0
votes
3answers
51 views

Logarithmic function and $x$

Hi I was wondering why in a logarithm $x$ cannot be a negative number, since for the inverse graph I drew the $x$ values are only positive. In the question it asks why the first four points of the ...
0
votes
1answer
55 views

Determine equation from graph

Background: I'm working on a script to read/parse a file generated by a piece of software I use to create music mixes. One aspect I'm having difficulty with is translating the volume value from it's ...
0
votes
1answer
191 views

Derivatives and Integrals of Polynomials and more.

I noticed that if I had a function $f(x)=x^n$ where $n$ is an integer, then $\lim_{m\to{n^+}}f^{(m)}(x)=n!$ where $f^{(m)}(x)$ is the $m$-th derivative. Also, ...
3
votes
5answers
1k views

Integrate by parts: $\int \ln (2x + 1) \, dx$

$$\eqalign{ & \int \ln (2x + 1) \, dx \cr & u = \ln (2x + 1) \cr & v = x \cr & {du \over dx} = {2 \over 2x + 1} \cr & {dv \over dx} = 1 \cr & \int \ln (2x ...
0
votes
4answers
47 views

Limit with logarithms

Can you please help to calculate the following limit $$\lim_{x \to \infty} \left(\frac{\ln(x\ln(\gamma) )}{\ln(x\ln(\gamma)+\ln(\frac{\gamma-1}{\gamma}))}\right)^{1/2},$$ where $\gamma >10$ is a ...
0
votes
2answers
50 views

Solving for $x$ using $\ln$

I have an equation as follows: $\ln(a)x + \ln(b) (1 - x) = \ln(c)$ I'd like to solve for $x$, but I think what I've come up with is wrong $\ln\left(\frac{a}{b}\right)x + \ln(b) = \ln(c)$ ...
0
votes
2answers
41 views

Solve simultaneous equations $\log(x-2)+ \log 2=2 \log y$, $\log (x-3y+3)=0$ (Not sure of solutions in book)

Solve simultaneous equations $\log(x-2)+\log2=2\log y$, $\log(x-3y+3)=0$ (Not sure of solutions in book) My method: $\log(x-2)+\log 2-\log y^2=0 \Rightarrow \log\left(\frac{x}{y^2}\right)=0 ...
1
vote
2answers
77 views

Is the complex square root of $z^2 = \pm z$?

Is $\sqrt{z^2} = \pm z$, for $z$ complex? I think it is, since either $-z$ or $+z$ satisfies the definition $\sqrt{z^2}= e^{\large \frac{1}{2}\log(z)^2}$ but I just wanted to make sure. It's a bit ...
2
votes
3answers
74 views

How do I evaluate the sum $\sum_{k=1}^\infty\left(\ln\big(1+\frac{1}{k+a}\right)-\ln\left(1+\frac{1}{k+b}\big)\right)$ [closed]

How do I evaluate the sum $$\sum_{k=1}^\infty\left(\ln\Big(1+\frac{1}{k+a} \Big)-\ln\Big(1+\frac{1}{k+b}\Big)\right)$$ where $0 <a<b<1$? Hints will be appreciated Thanks
1
vote
0answers
35 views

On the dog-bone contour around [-1,1], what are the arguments of these two lines approaching the real axis from above and below?

I am using a dog-bone contour to integrate around the interval [-1,1]. (-1 and +1 are branch points of the integrand.) I am using the principal branch of log, so I am restricting its argument to ...
4
votes
1answer
80 views

Summation of series containing logarithm: $\sum_{n=1}^\infty \ln \frac{(n+1)(3n+1)}{n(3n+4)}$

How do I find the sum of the series: $$\ln \frac{1}{4} + \sum_{n=1}^\infty \ln \frac{(n+1)(3n+1)}{n(3n+4)} $$ I tried expanding the terms on numerator and denominator and got $$\ln \frac{1}{4} + ...
0
votes
1answer
18 views

Given $10^3=1000,10^4=10000,2^{10}=1024,2^{11}=2048,2^{12}=4096,2^{13}=8192$,what are the largest $a$and smallest $b$ such that $a < \log_{10} 2 < b$

If one uses only the information $10^3=1000,10^4=10000,2^{10}=1024,2^{11}=2048,2^{12}=4096,2^{13}=8192$,what are the largest $a$ and smallest $b$ such that one can prove $a < \log_{10} 2 < ...
2
votes
1answer
63 views

Name for some kind of logarithmic norm/error

As known $(\mathbb R, +)$ and $(\mathbb R^{+}, \cdot)$ are isomorphic with $\exp:\mathbb R\to\mathbb R^{+}$ as an isomorphism. When I transfer the absolute value $|\cdot|$ on $(\mathbb R, +)$ via ...
0
votes
1answer
37 views

Domain of the nested logarithmic function

Find the domain of the definition of the function $$f(x)=\log_{0.3}\left(\log_{0.5}\left(\log_{0.8}\left(x^2-x+1\right)\right)\right)$$ My Try: I assumed $$f_1(x)=x^2-x+1$$ ...
3
votes
0answers
33 views

Branch points and Riemann surfaces (analytic continuation),

Take probably the most typical example: $$f(z) = \sqrt{1-z^2}$$ This function uses the (complex) logarithm to define it: $$e^{\large \frac{1}{2}log(1-z^2)}$$ $$e^{\large \frac{1}{2}[ln|1-z^2| + ...
7
votes
2answers
85 views

Solving a logarithmic equation $\log_2 (2^x-1)+x=\log_4 (144)$

I need to solve this: $$\log_2 (2^x-1)+x=\log_4 (144)$$ I can work out that $x=\log_2 (2^x)$ and $\log_4 (144)=log_2(12)$ but I'm stuck after that.
2
votes
3answers
78 views

For which values of $a\in\mathbb{R} $ the equation $2 \log(x+3)=\log(ax)$ has exactly one root?

I have to investigate the possible roots of the equation according to $a$, i.e. i have to see whether there is only one root, two roots, or no roots and also what their sign is each time. This is from ...
2
votes
3answers
39 views

Divergence of $\sum_{n\geq 2} \frac{1}{\ln^p n}$ for $1<p\leq \infty$ [closed]

Can anyone help me to prove that $(x_n)\notin l_p$ with $x_n=\frac{1}{\ln^p n}$? Suppose $1< p<\infty$.
71
votes
6answers
3k views

Prove $\left(\dfrac{2}{5}\right)^{\frac{2}{5}}<\ln{2}$

Inadvertently, I find this interesting inequality,But this problem have nice solution? prove that $$\ln{2}>(\dfrac{2}{5})^{\frac{2}{5}}$$ This problem have nice solution? Thank you. ago,I find ...
0
votes
0answers
47 views

Calculation of infinite product

My question is to prove the identity: $$ \prod_{n=1}^{\infty}\left(\frac{\cos t-1}{n}+1\right)=\exp\left(-\int_0^1x^{-1}(1-\cos xt)dx\right) $$ which arises as a product of characteristic functions of ...
4
votes
5answers
338 views

How to find the limit $\lim \limits_ {x \to+\infty} \left [ \frac{4 \ln(x+1)}{x}\right]$

Solve $\space \begin{align*} \lim_ {x \to+\infty} \left [ \frac{4 \ln(x+1)}{x}\right] \end{align*}$. I did this way: $$\begin{align*} \lim_ {x \to+\infty} \left [ \frac{4 \ln(x+1)}{x}\right] ...
-1
votes
3answers
2k views

Upper Bound of Logarithm

For $1\leq x < \infty$, we know $\ln x$ can be bounded as following: $\ln x \leq \frac{x-1}{\sqrt{x}}$. Then what is the upper bound of $\ln x$ for following condition? $2\leq x <\infty$
13
votes
5answers
472 views

How to evaluate $\int_0^1 (\arctan x)^2 \ln(\frac{1+x^2}{2x^2}) dx$

Evaluate $$ \int_{0}^{1} \arctan^{2}\left(\, x\,\right) \ln\left(\, 1 + x^{2} \over 2x^{2}\,\right)\,{\rm d}x $$ I substituted $x \equiv \tan\left(\,\theta\,\right)$ and got $$ ...
1
vote
1answer
33 views

How to derive this simple equality?

Let us define $L_i\triangleq \log \left( \dfrac{Prob(x_i=+1) }{ Prob(x_i=-1)} \right)$ $E\{x_i\} \triangleq Prob(x_i=+1)-Prob(x_i=-1)$ I need to show that \begin{equation} E\{x_i\} = ...
8
votes
5answers
6k views

Approximating Logs and Antilogs by hand

I have read through questions like Calculate logarithms by hand and and a section of the Feynman Lecture series which talks about calculation of logarithms. I have recognized neither of them useful ...
7
votes
3answers
137 views

Finding $\int_0^{\pi/4}\sqrt{1+\left( \tan x\right)^2}dx$

I would like to understand all the steps to find out this integral $$ \int_0^{\pi/4} \sqrt{1+\left( \tan x\right)^2} dx$$ Wolfram Alpha returns: $$ \frac12 \log(3+2 \sqrt2) = 0.881373587019543...$$ ...
6
votes
2answers
145 views

Does logging infinitely converge?

Trying to evaluate $$\ln(\ln(\ln(\ln(\cdots\ln(x)\cdots))))$$For some fixed $x$ produces a complex answer that appears to converge, at least sometimes. So I want a proof that this converges for ...
2
votes
1answer
79 views

$c= (a^{x}-b^{x})$ where $a$,$b$ and $c$ are known real constants. Solve for $x$.

I tried taking $\log$ on both side but i ended with $\log(a^{x}-b^{x})$ which is difficult to solve. Does anybody has idea how to solve the above equation for $x$.
0
votes
2answers
37 views

If $f(x)=\log \left(\cfrac{1+x}{1-x}\right)$ for $-1 < x < 1$,then find $f \left(\cfrac{3x+x^3}{1+3x^2}\right)$ in terms of $f(x)$.

If $f(x)=\log \left(\cfrac{1+x}{1-x}\right)$ for $-1 < x < 1$,then find $f \left(\cfrac{3x+x^3}{1+3x^2}\right)$ in terms of $f(x)$. My Attempt $$f ...
1
vote
0answers
15 views

Multiplicative & additive measurement error models concerning logarithms

I understand that taking the logarithm of the multiplicative error model transforms it into the additive error model. Let $y'$ be the observed response variable, with $y$ being the true response ...