Questions related to real and complex logarithms.

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0
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2answers
29 views

Help to find the best lower bound function for a given set of data, based in the natural logarithm function

I am trying to find a lower bound function for a set of data I have, and I am struggling with it. In the following graph the blue color is the set of data and the red color is my lower bound function. ...
1
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1answer
16 views

The position of significant digits and Logarithms relationship…

I am unable to solve the following question has i don't understand what the relationship is between significant figures and Logarithms. Q-If $\log_{10}(7)= 0.8451$ then the position of the first ...
2
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1answer
29 views

Stuck with understanding transformation step in calculating limit of $n(\sqrt[n]{a}-1)$

Although this question has already been asked in general ( $\lim\limits_{n\to\infty} n·(\sqrt[n]{a}-1)$) , my question is different, because I am stuck with a specific transformation step: ...
-1
votes
2answers
33 views

Integration of logarithm

$\int \ln(\ln \sqrt{x})^{\ln (x)}dx$ how should I integrate this? I think it can't be integrated. I don't know.
1
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1answer
49 views

Fourier transform and splitting frequency range into 4 channels

I have code example that divides audio frequency into 6 channels. It uses Fast Fourier Transform (FFT). Algorithm process the frequency range using 6 capture[x] samples based on the range of n between ...
0
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2answers
51 views

What is this equation?

I ran across this equation for use in web code here and am desperately wanting to know if any portion of it or the whole thing is a standard equation somewhere. This is the best I could do ...
0
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2answers
36 views

rewrite logarithmic expression

I have this logarithmic expression 2 logb 6 + (1/2) logb 25 - logb 30 and have to rewrite it as logb of one number. I just don't understand how to do this. help please.
1
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1answer
32 views

How to prove that $f(x) = x^ε - \log x$ is $\infty$ when $x\to\infty$?

I'm trying to prove that the function $x^ε$ is "bigger" than $\log x$ when $x\to\infty$, for every $ε>0$. Or to put it in a more formal way: For every $ε>0$, there exists a constant $N$ for ...
0
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1answer
32 views

Simplification of a logarithm expression

I need to verify the answer of a logarithm expression (note, I'm not a student). I managed to get through high school and college without ever having a math course that taught logarithms--I don't ...
0
votes
1answer
20 views

Deriving a function with logarithmic terms

Let $L(X) = \exp(\sqrt{\log X \log \log X})$ Prove that if $c > 0$,$ Y = L(X)^c$, and $u = \log X/ \log Y$ , then $$u^u = L(X)^{(1/2c)(1+o(1))}$$ I've tried to write $u^u = (\log X/ \log ...
1
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1answer
30 views

Logarithm, Just need help understanding what this question is asking. Not looking for an answer.

In my foundations of computing class, we were given a logarithm question which i don't quite understand. This is the question. Given the logarithmic table values of the numbers x and y are ax and ay ...
1
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2answers
50 views

Deriving properties of the logarithm from its integral representation

Suppose we define: $$\ln(x) = \int_{a}^{x} \left[ \frac{1}{r} dr\right]$$ Such that $$ \ln(1) = 0, \ln(e) = 1$$ How does one derive all the properties of the logarithm from the properties of the ...
1
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1answer
29 views

proving with a sequence

The question is : Show that if $n$ is a power of $2$, then $$\sum_{i=0}^{\log_2n-1}2^i=n-1\;.$$ Tried induction at first and tried to prove it on 2n but nothing came out of it. Then i tried ...
0
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3answers
29 views

if $x>1$ and $\log_2x,\log_3x,\log_x16$ are in G.P then what is x $=$

if $x>1$ and $\log_2x,\log_3x,\log_x16$ are in Geometric progression then what is x equal to? Solution: $(\log_3x)^2=\log_2x\times\log_x16=\log_216=\log_22^4=4$ $\log_3x=2 or x=3^2=9$ so my doubt ...
-4
votes
1answer
75 views

Check whether a function is one-to-one and onto

If $f(x) = \log_{x^3}\left(\sqrt{x}\right)$, check whether $f$ is one-to-one and onto where $x\in R^+\setminus\{1\}$. Also write the range of $f$. Alright, if $f(m) = f(n)$ and if we would prove m=n ...
0
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6answers
62 views

Limit of log functions

I need help solving this problem. $$\displaystyle \lim _{x\to 0}\frac{\log\left(1+7x\right)}{5x}$$
1
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3answers
38 views

Operations and Identities [duplicate]

We have the binary operation addition on numbers. It has an additive identity ( 0 ) and it is commutative. Multiplication is simply repeated addition. It is a binary operation on numbers. Its ...
1
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3answers
40 views

How do you solve this logarithmic equation?

While reading through my textbook, I came across this particular equation: $$ x = x\log (y) + \log (y) $$ But they solve it by doing this: $$ x = x\log (y) + \log (y) $$ $$ x = (x + 1)\log(y) $$ $$ ...
0
votes
3answers
18 views

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
votes
2answers
49 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 ...
0
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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 ...
0
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1answer
43 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 ...
0
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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 ...
2
votes
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...
0
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1answer
33 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
votes
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 ...
0
votes
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 ...
1
vote
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 ...
2
votes
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 ...
0
votes
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
votes
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
votes
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.
0
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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
votes
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} ...
1
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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 ...
0
votes
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
votes
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, ...
3
votes
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
votes
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 ...
0
votes
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
votes
4answers
281 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
votes
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
votes
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
votes
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 ...
1
vote
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
votes
1answer
33 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
votes
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 ...