Questions related to real and complex logarithms.

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2answers
265 views

For what $n$ does $[\log_21]+[\log_22]+[\log_23]+\dotsb+[\log_2n] = 1538$? [duplicate]

I just can't solve this problem in spite of doing a whole book on logs and inequalities Where $[\dotsc]$ denotes the greatest integer function, what is the value of the natural number $n$ ...
0
votes
2answers
50 views

Tricky Logarithmic Inequality Problem

I am having a problem solving this question - If $\log_{\frac{1}{\sqrt{2}} }{\sin{x}}>0$, $x\in [0,4\pi]$,then number of values for chating which are integral multiples of $\pi/4$,is A-6 B-12 ...
2
votes
3answers
77 views

Integral of $\log(\sin(x)) \tan(x)$

I would like to see a direct proof of the integral $$\int_0^{\pi/2} \log(\sin(x)) \tan(x) \, \mathrm{d}x = -\frac{\pi^2}{24}.$$ I arrived at this integral while trying different ways to evaluate ...
1
vote
0answers
25 views

Taking integral of the complex logarithm using fundamental theorem?

Is it valid to do this? I have $f(z)= z^i$,and $F(z)=\frac{z^{i+1}}{i+1}$ and assuming we're using principle values of $f$ and $F$ would it be correct to say that: $\int_{-1}^{1} f(z) dz = ...
2
votes
0answers
26 views

Asymptotic solution to $m \leqslant e^{\lambda t} (c t^q - \varepsilon)$

What is the smallest $t$ statisfying the inequality: $m \leqslant e^{\lambda t} (c t^q - \varepsilon)$, where $\varepsilon$ is arbitrary small positive number? I believe $t$ must be of the from: $$t = ...
0
votes
1answer
43 views

Logarithmic question

In the following question I fail to understand why the A option is correct. I understand that D is wrong, and that B and C are correct, but why is A correct? If $3^x=4^{x-1}$, then $x $cannot be ...
4
votes
1answer
48 views

Integration of Exponential and Logarithms, $\int_{z-1}^z \log(\frac{1}{z-y}) \exp (-| y| ^{3}) \, dy$

The integral I am dealing with is: $$\frac{3}{2 \Gamma \left(\frac{1}{3}\right)}\int_{z-1}^z \log \left(\frac{1}{z-y}\right) \exp \left(-\left| y\right| ^{3}\right) \, dy$$ where $z\in \mathbb{R}$ ...
1
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2answers
40 views

Determine the convergence or divergence of $\sum_{2}^{\infty}\frac{1}{(\log n)^{s}}$, where $s \in \mathbb{R}$ is given.

Since $$\frac{1}{(\log n)^{s}} > \frac{1}{n^{s}}$$ for large $n$, if $s \leq 1$ then $\sum_{2}^{\infty}\frac{1}{(\log n)^{s}}$ diverges. But for $s > 1$ I have not yet figured out a proof.
0
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0answers
34 views

Taking the logarithm of a periodic function

I've been wondering how we take the logarithm of a periodic function. At least I think that's what I've been wondering - but I may have confused the terminology. Anyway, take, for example, the ...
0
votes
1answer
36 views

Minimum value of a Logarithmic equation

What is the minimum value of $$\log_a(x)+ \log_x(x) $$ where $0\leq a\leq x.$ I do not understand why my book says the answer is $2$ because when i take $a=0.1$ say and $x =0.2$ I get $\approx ...
0
votes
2answers
51 views

Is it true that $\int_{0}^{1}(1+x^{2})^{-1/2} = \log (1 + \sqrt{2})$?

Since $$D^{-1} (1 + x^{2})^{-1/2} = \sinh ^{-1} (x) + C,$$ is it true that $$\sinh ^{-1} x + C \big|_{0}^{1} = \log (1 + \sqrt{2})?$$ What relates $\sinh^{-1}(\cdot )$ to $\log(\cdot )$? Here ...
0
votes
2answers
35 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
vote
1answer
19 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
votes
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
34 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
vote
1answer
51 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
votes
2answers
54 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
votes
2answers
41 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
vote
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
33 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
34 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 ...
-3
votes
1answer
84 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
63 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
42 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
vote
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
19 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
52 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
votes
1answer
34 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
votes
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
41 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
61 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
vote
1answer
30 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
38 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
62 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
65 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
347 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
30 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
vote
2answers
29 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, ...