Questions related to real and complex logarithms.

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Derivate a logaritmic function

Let's take $ f = \ln(x) $. The derivate is $ f' = 1/x$. However $g = \ln(50x) $ has the same derivate $f' = g'$. How come? If I where going to derivate $g$ I would substitute $x$ for $t$: $g = ...
0
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2answers
48 views

Is $\ln n$ transcendental for all rational $n>1$?

I know that $\ln n$ is transcendental for all integer $n>1$. But does this still hold for non-integer rational values of $n>1$? For example, is $\ln 1.5$ transcendental? EDIT: Somehow managed ...
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1answer
32 views

logarithmic differentiation issue

Trying to understand a solution I was given to a problem I was told to use logarithmic differentiation on. $$ 1/x(x+1)(x+2) $$ and I know that $$log((ab)/c) = log(a) + log(b) - log(c)$$ So I tried to ...
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1answer
42 views

Implications of redefining base natural logarithm constant e

Disclaimer: I'm no math expert! I understand that the constant $$e$$ is expressed as follows: $$e = \sum_{n=0}^{\infty} \frac1{n!} = 1 + \frac1{1*1} + \frac1{1*2} + ...$$ What would be the ...
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1answer
29 views

Properties of a different kind of a logarithm

We all have heard about the natural logarithm for any number. Basically we all know that the natural logarithm is the logarithm to the base of $e$,which is a transcendental number. Now what about the ...
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2answers
39 views

Properties of natural logarithm

$\ln( n + 1 ) - \ln( n ) > \frac 1{n+1}$ Is this statement true? I tried to show by $$\ln( n+1 /n)\implies 1+ 1/n > 0, \quad n >1$$ That is all I could get to so...
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1answer
31 views

Find a branch of $\log (2z - 1)$ that is analytic at all points in the plane

Find a branch of $\log (2z - 1)$ that is analytic at all points in the plane except those on the following rays a) {$x + iy : x \leq \frac{1}{2}, y = 0$} Definition: $F(z)$ is said to be a branch ...
3
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2answers
60 views

Logarithms and ratios.

This is the question: $$\log_b 64 = \frac{3}{b}$$ And have to find $b$. So I tried a bit and got this:$$\frac{b}{\log b} = \frac{\log 64}{3}$$ But have no idea what to do next. Thanks for your ...
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0answers
70 views

Fourier transform of $ \log(x^{2}+a^{2}) $

I would like to evaluate the Fourier cosine transform of $\log(x^{2}+a^{2})$ or the integral $$\int_{0}^{\infty}\cos(ux)\log(x^{2}+a^{2})\,dx$$ for any real $u,a$. However, it seems that this ...
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1answer
22 views

re-arrange equation $L=2^{10(v-1)} v^2$

Is it possible to re-arrange this equation to make v the subject? $$L=v^2 . 2^{10(v-1)}$$ If so, what is the answer? If it helps (which by excluding zero it should)... $$0<v<1$$ I have tried ...
1
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1answer
27 views

Practical use for non-integer logarithmic bases

Are there practical uses (ie: in engineering, applied sciences, chemistry, IT, etc) for using non-integer bases? From other questions on the topic, I see that it's just another way of representing ...
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2answers
75 views

$\lim_{x\to1}\frac{\ln(x)}{x-1}$ and its strange graph

I was studying exponential growth and noticed that $\ln(0.99) \approx −0.010050336$ $\ln(0.999) \approx −0.0010005$ $\ln(0.9999) \approx −0.0001 \ldots$, and also $\ln(0.9952) \approx −0.004811557$ ...
1
vote
1answer
30 views

What is a simple way of describing branch cuts?

Branch cuts have been asked about and discussed on MSE extensively. That is, every answer to something along the lines of "What is a branch cut?" is... extensive. I'm looking for a quick, intuitive ...
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1answer
61 views

Approximation for $ e^{ - x^2 } $ , x>0.

what is the good approximate so that it works for a large range of values. My purpose is to calculate logarithm of likelihood ratios. $ \log \left( {\frac{{e^{ - x_1 ^2 } + e^{ - x_3 ^2 } }} {{e^{ - ...
0
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0answers
55 views

Beyond taylor series?

Consider functions $f(z)$ that are analytic for $Re(z) > 0$ and are also analytic for $(Im(z))^2 > 0$. Let $n$ be a nonnegative integer. Now I define some series expansion of "order $n$" , ...
6
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3answers
339 views

Definite integral involving logarithm of cosine

Does anyone know the provenance of or the answer to the following integral $$\int_0^\infty\ \frac{\ln|\cos(x)|}{x^2} dx $$ Thanks.
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1answer
18 views

Best way to prove all 3 solutions for exponential equation?

I was given the equation; $(x-7)^a=1$ where $a=(x-4)$ The 3 solutions are: $x=4, 6, 8$ When $x=4$, $(-3)^0=1$, which can be reached by setting $(x-4)=0$ because $n^0=1$ When $x=8$, $1^4=1$, ...
0
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2answers
29 views

Implicit logarithmic differentiation to find the horizontal tangents of an exponential function

The graph of $y = 6{(3{x}^2)}^x$ has two horizontal tangent lines. Find equations for both of them. $$ \\ \begin{align} \\ y &= 6{(3{x}^2)}^x \\ y &= 6 \cdot {3}^x \cdot {x}^{2x} \\ \ln{y} ...
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2answers
28 views

Problems with this max likelihood estimation

I have the following density function: $f(x;\omega) = \omega*x^{(\omega-1)}*I_{(0,1)}(x)$ for $\omega > 0$ First I want to have the Likelihoodfuntion, which is $\prod_{i=1}^n f(x_i;\omega)$ I ...
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1answer
33 views

finding the methodology of solving logarithmic equation

Find the value of $\log_{3} (3^{2x}-3^x+1) = x$. How should we get the value of $x$. $x$ is equal to $0$ but problematically I can't find a way to show that.
0
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1answer
49 views

When do we have $Dx^{r} = rx^{r-1}$ for $x \leq 0$?

Since, if $x > 0$ then $Dx^{r} = rx^{r-1}$ for real $r$, when do we have this result for $x \leq 0$? I think the point is to circumvent the trouble that if $x \leq 0$ then $\log x$ is meaningless, ...
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1answer
70 views

What is a branch cut? [duplicate]

This may be a strange question; but I've read and re-read the chapter in my textbook on what exactly a branch of a logarithm is and am having trouble understanding. What is a branch of a ...
2
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1answer
43 views

Are there exception cases when you are bringing an exponent out of a logarithm?

The domain of a logarithm $\log(x^2)$ is $D:x\in(-\infty,0)\cup(0,\infty)$. But if I use the identity $\log(a^b)=b\log(a)$ and do: $\log(x^2)=2\log(x)$ the domain becomes $D: x\in(0,\infty)$ The ...
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1answer
30 views

Help solving non-trivial logarithmic inequality

I have the following equation: $$\dfrac{2\pi G\lambda M^4}{m^2}\ln\left(\dfrac{\phi_i}{\phi_e}\right)+2\pi G\left(\phi_i^2-\phi_e^2\right)\ge 65$$ which, for the purpose of this question, I'll ...
4
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4answers
273 views

The caret ^ symbol means exponentiation informally in math. Why not a symbol for log too? [closed]

Plus, minus, multiply, divide, and exponentiation all have symbols in math (+, -, *, /, ^ ) . But why isn't there the missing log symbol too? Here's how it would work: 4 ^ 5 = 1024 (as is standard ...
2
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2answers
152 views

Approximating $\ln(1+\exp(x)+\exp(y))$

Ok, so I know that $\ln(1+e^x)\approx x$ when $x$ is large. But what about $\ln(1+e^x+e^y)$ when both $x$ and $y$ are large? I can figure out cases when $x\gg y$ or $y\gg x$ since that simplifies ...
0
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1answer
96 views

The value of the logarithmic expression can never be $\ldots$

The value of the logarithmic expression $\log_x \dfrac{x}{y} +\log_y \dfrac{y}{x},\text{where}\quad x\geq y>1\quad$ can never be $\bf\text{options}$ a.) $-1\quad$ b.)$\quad0.5\quad$ c.) ...
2
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1answer
24 views

Is there any way to extend the domain of this function through analytic continuation?

$\prod_{k=2}^x \log k=F(x)$ It looks a lot like the gamma function (a sort of logarithmic factorial), and I wonder if it can be similarly expressed as an integral or something. Any ideas?
0
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1answer
16 views

How to solve this Diophantine equation (involving natural logarithms)?

The equation is $r = \ln{a} + b \ln{c}$ where $r \in \mathbb{R}$ is fixed and $a,b,c \in \mathbb{N}$. In other words, for arbitrary real r, how can one say whether a solution (in form above) exists ...
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2answers
38 views

How can I isolate for the $z$ exponent?

Can anyone help me with this math equation? Solve for $z$ $$P = \frac{e^z}{1 + e^z}$$ $$P(1 + e^z) = e^z$$ $$P + Pe^z = e^z$$ $$P = e^z - Pe^z$$ I've got this far, am I at least on the ...
0
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1answer
32 views

Need help with logarithmic differentiation

I need to use logarithmic differentiation to get f(x)=x$\sqrt{(x+1)(x+2)(x+3)(x+4)}$. I've been working on it for a while and could use some help. Thanks!
3
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2answers
56 views

Compute Power Series Convergence to a function

Consider the next power series $$ \sum_{n=1}^{\infty} \ln (n) z^n $$ Find the convergence radius and a the function $f$ to which the series converges. I have easily found that $R=1$ is the ...
0
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2answers
75 views

Solve $2^x=13 \bmod 3^4$

Solve $2^x=13\bmod 3^4$ I know $\log13=30\bmod 3^4$ and $\log16=15 \bmod 3^4 $ I've tried subbing $\log13/\log16$ for $2$ but I am not sure what to do next.
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0answers
27 views

Are Percentile and Logarithm Exchangable?

I am wondering if the Percentile Operator, denoted as 'prctile', commutates with the logarithm of basis 10, denoted as 'log10'. Is the following statement true, and if yes, why? log10(prctile(X)) = ...
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0answers
24 views

Need help to Compute a specific derivation

I have a function equal to: Where L is an orthonormal matrix of eigenvectors of a matrix S, and ...
2
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1answer
60 views

How to solve this logarithmic inequality?

I've started a data structure course and I need some help with solving these logarithmic inequalities. It would also be helpful because later on these kind of calculation won't pose a problem later ...
1
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1answer
43 views

Could someone explain steps?

I am learining about logarithm equations, and i can´t seem to understand how to solve such an equation, could someone help? I must solve the equation/find $x$ for: $$2^{2x} - 3\cdot2^x - 10=0$$ The ...
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2answers
60 views

Find $\lim\limits_{n \to \infty} \frac{\log(1+2^n)}{\log(1+3^n)}$

How to calculate this limit? $$\lim\limits_{n \to \infty} \frac{\log(1+2^n)}{\log(1+3^n)}$$
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3answers
39 views

Question regarding logarithms 2

What is $\ln(-1)$? And would there a taylor series for $$\ln\frac{1+x^m}{1-x^m}$$?
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1answer
12 views

Solve for r. Logarithms

$$ 36000 = 3450 * \frac{1-[1/(1+r)^{12}]}{r} $$ The next step is divide both sides by 3450. Now I'm stuck. Help solve for r.
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2answers
79 views

Integration of 1/x as a limit of a sum

This is from R.Courant book Example "Introduction to Calculus and Analysis vol.1 " To integrate $x^\alpha$ when $\alpha\neq1$ we subdivide the interval [a,b] by the point of geometric progression: ...
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3answers
38 views

Question regarding logarithms

Can you factor out the $m$ out of $\ln(c\cdot x^m)$ where $c$ is a constant?
0
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1answer
49 views

Logarithm inequality for specific range

I need to show that: $$ \ln(1+x)\left(\ln\left(\frac{1+x}{1-x}\right)+1\right)+\ln(1-x)\ge 0, $$ for $0\le x\le 2/3$. Thanks
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0answers
13 views

Calculating gain ratio from a dB value

In a practice problem I have: power gain = $\log_{10}(\frac{db}{20})$ The final answer for the ratio is 1. The dB value is $-3$. When I do $\log_{10}(\frac{3}{20})$ I get $-0.823$. Just wondering ...
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1answer
42 views

Why is $\frac{\sum_{i=1}^n \log(X_i)}{n} = \overline{log X}$ [closed]

Why is $$\frac{\sum_{i=1}^n \log(X_i)}{n} = \overline{log X}$$ ($X_i$ are i.i.d samples)
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2answers
56 views

Why does the log-normal probability density function have that extra “x”?

For a random variable $X \sim N(\mu, \sigma^2)$, the probability density function is $$f(x; \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} \cdot \exp\left\{ -\frac{(x-\mu)^2}{2\sigma^2} \right\}$$ ...
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1answer
44 views

How do we solve for $n$?

Asymptotic complexity gives an idea of how rapidly the space/time requirements grow as problem size increases. • Suppose we have a computing device that can execute 1000 complex operations per ...
3
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1answer
94 views

Log or Antilog tables, which ones are more useful?

I'm trying to make a Log or Antilog table small enough to fit in the back of a wallet calendar (or a business card). My intend is to build a mathematically useful gift that can be used by anybody ...
4
votes
3answers
81 views

Closed form for the partial sum $\sum\limits_{k = 1}^n \frac{\ln k}k$

I'd like to find a closed form for this partial sum: $$\sum\limits_{k = 1}^n \frac{\ln k}k$$ Using the properties of the logarithms, I converted the above into $$\ln\left(\prod_{k = 1}^n ...
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4answers
64 views

Logarithm and trigonometry

Is $\ln (\sin x-\cos x)$ equal to $\ln (\cos x-\sin x)$? So I did a integral problem but the answer is not same the answer given. I'm given this question $\int (\frac{2}{1-\tan x})dx$ So I got ...