Questions related to real and complex logarithms.

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0answers
23 views

1st order ODE separable

everyone! :-) I've a ODE question with I can't solve. It's here: ${dy\over dx} = {{xy + 2y-x-2}\over {xy-3y+x-3}} $ I tried the following: ${dy\over dx} = {{xy + 2y-x-2}\over ...
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3answers
36 views

How is this logarithmic identical transformation true? [closed]

$$x^{1-\log x}=1\Leftrightarrow(1-\log x)\log x=\log 1?$$ I don't know how it can be true.
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1answer
22 views

How is this identical transformation true $x^{1-\log(x)}=1\Longleftrightarrow \log x^{1-\log(x)}=\log1$?

How is this identical transformation true $$x^{1-\log x}=1\Longleftrightarrow\log x^{1-\log x}=\log1\text{ ?}$$ I thought to put both sides on log: $$\log x^{1-\log x}=\log1,$$ but then I don't know ...
0
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1answer
44 views

Evaluation of Spence's function.

Spence's function is defined as $${\rm Li}_2 (z)=- \int_0^z \frac{\ln(1-u)}{u} \, du $$ where $$z \in {\mathbb C} \setminus [1, \infty )$$ For $|z|<1 $ $${\rm Li}_2 (z)= \sum_1^ \infty \frac{ ...
0
votes
2answers
27 views

How to get the graph for $y= \log_{1/a} (x)$ from $y= \log_a (x)$?

I know that it is symmetric the Ox axle, but I can't prove it.
1
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3answers
40 views

Limit of $y \ln (x^2+y^2)$

I want to calculate limit of $\lim_{(x,y) \rightarrow(0,0)}y \ln (x^2+y^2)$. How to do that? From iterated limits i know that limit exists for certain, but how to show that it is equal to zero then?
0
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1answer
15 views

If the positive numbers x,y,z are in harmonic progression, then log(x+z) + log(x-2y+z) equals

If the positive numbers x,y,z are in harmonic progression, then log(x+z) + log(x-2y+z) equals a) 4log(x-z) b) 3log(x-z) c) 2log(x-z) d) log(x-z) How do i approach this problem? IF x,y,z are in HP, ...
1
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5answers
97 views

How to solve $\ln(x) = 2x$

I know this question might be an easy one. but it has been so long since I solved such questions and I didn't find a an explanation on the internet. I'd like if someone can remind me. I reached that ...
0
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1answer
22 views

Number of integral values of M

If $\log_3M=a_1+b_1$ and $\log_5 M=a_2+b_2$, where $a_1,a_2$ are natural numbers and $b_1,b_2 \in [0,1)$. If $a_1a_2=6$, then find the number of integral values of M. What so I do in the problem. I ...
3
votes
1answer
41 views

System of logarithmic equations

$$\log (2000xy)-\log x\log y=4$$$$\log(2yz)-\log y\log z=1$$$$\log(zx)-\log z\log x=0$$ The base is 10 everywhere. I tried opening the log with the sum formulae and then manipulating, but I got stuck. ...
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1answer
25 views

How to find the domain of the function $f(x) =\log_y a^2$?

To find the domain of the function $f(x) = \log_y a$ it's enough to check if the base (y) is greater than 0 and not equals to 1 and the number (a) is greater than 0. But what if we have a power of ...
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0answers
11 views

GLM for normal distribution

$Y_i$~N$\left(\mu_{i,}\sigma^2\right)\space \mu^2=\alpha+\log \left(\beta_0+\beta_1x_i\right)\space \alpha\space is\space unkown$ how is this proved to be a Genralized Linear Model? My assumption ...
10
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4answers
178 views

Calculate the infinite sum $\sum_{1}^\infty \frac{\log{n}}{2n-1}$

I would like to calculate an asymptotic expansion for the following infinite sum: $$\displaystyle \sum_{1}^N \frac{\log{n}}{2n-1}$$ when $N$ tends to $\infty$. I found that the asymptotic expansion ...
2
votes
3answers
100 views

How could this be true $n=\log(e^n)$?

I am learning elementary logarithms. How could this be true $n=\log(e^n)$? I searched online and couldn't find any info on this, could anyone give me some clue?
0
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2answers
38 views

Even and odd functions | logarithm [closed]

show why this logarithm is an odd function? $$y = \log_2 (x-\sqrt{1-x^2})$$
0
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7answers
104 views

Given $\log 2$ and $\log 3$, compute $\log 120$ [closed]

Given that $\log2= 0.30103$ and $\log3=0.47712$, calculate (without using tables or calculators) the value of $\log 120$.
3
votes
1answer
56 views

Find integer $n$ that satisfies $(\lg n)^{2^{100}} <\sqrt{n}$ with $n > 2$

If $(\lg n)^{2^{100}} < {n^{1/2}}$, where $\lg$ is the binary logarithm, then $$(\lg n)^{2^{101}} < n$$ $$2^{101}\lg \lg n < \lg n$$ $$101 < \lg \lg n - \lg \lg \lg n$$ I don't know that ...
0
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2answers
25 views

Show $\\Log z_1z_2 \neq Log z_1 + Log z_2$. given $z_1 = i$ and $z_2 = -\sqrt 3 + i$.

Show by evaluating both sides that for $z_1 = i$ and $z_2 = -\sqrt 3 + i$, $\\Log z_1z_2 \neq Log z_1 + Log z_2$. Recall the definition: $\\Log z = Log |z| + iArg z$ Attempt: left side: $\\Log ...
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3answers
63 views

Solve for X in a difficult exponential function [closed]

Solve for $X$ when $3^{x^x}=1000$ By hand please (without evaluating the intersection on the graph). How is it done?
5
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1answer
142 views

Why are logarithms of trigonometric functions useful?

I have noticed that in many trigonometric tables the logarithm of the trigonometric values are given. Why this is given and not the actual values of the trigonometric functions? For example, instead ...
1
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0answers
37 views

Generator of group, Computation of discrete logarithm

The prime number $p=67$ is given. Show that $g=2$ is a generator of the group $\mathbb{Z}_p^{\star}$. Compute the discrete logarithm of $y=3$ as for the base $g$ with Shanks-algorithm. Compute the ...
2
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5answers
84 views

How to properly solve this inequality $2^x < \frac{3}{4}$?

How to properly solve this inequality? $$2^x < \frac{3}{4}$$ I know that it will be something like that: $$ x \stackrel{?}{\ldots} \log_2\frac{3}{4} $$ But I don't know how to decide if it should ...
1
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2answers
35 views

What are these numbers in a logarithmic table?

Below is an image from a table of logarithms. As an example, one sees that $\log(661.3) = 2.82\color{red}{040}$. In this logarithmic table there are some numbers to the right. My question is: What is ...
0
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2answers
43 views

Prove that if $x>1$, then $log_a(x)>0$

If $x>1$ then $log_a(x)>0$ Well I thought that log with base a of 1 is 0. I don't know what to do more.
1
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1answer
36 views

Is $1 + \lg i = \lg(i + i)$?

I've been studying Sedgewick's "Algorithms" book and in proof of one proposition he writes the following: the property is preserved because $1 + \lg i = \lg(i + i) \le \lg(i + j) = \lg k$ I ...
0
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4answers
107 views

Which is more preferable to write $\log(x)$ or $\ln(x)$ [duplicate]

Which one is more preferable to write when you are writing an exam. Is it $\log(x)$ which denotes the natural logarithm or is it $\ln(x).$
1
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2answers
62 views

Integration By Parts (Logarithm)

$$\int(2x+3)\ln (x)dx$$ My attempts, $$=\int(2x\ln (x)+3\ln (x))dx$$ $$=2\int x\ln (x)dx+3\int \ln(x)dx$$ For $x\ln (x)$, integrate by parts,then I got $$=x^2\ln (x)-\int (x) dx+3\int \ln(x)dx$$ ...
0
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1answer
47 views

Can I simplify $ \ln(A/B)+C$ any more?

This should be a rather simple problem however I am having difficulty getting this simplified. If I need to simplify the expression $$ \ln(A/B)+C$$ My first step is $$ A/B + e^c$$ However MATLAB and ...
0
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1answer
23 views

Factoring ln functions

can anyone tell me how the following factoring ends in $\ln x - \frac{ln 2}{2}$ Original $\frac{\ln x}{\ln 2} - \frac{\ln 2}{ln 2}$ Work shown from Professor $\frac{1}{ln 2} (\ln x - \frac{ln ...
0
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0answers
16 views

Show T(n)=T(n/5)+T(4n/5)+n/2 is $\Omega (n log n)$

I'm tasked with showing T(n)=T(n/5)+T(4n/5)+n/2 is Big-Omega n log n by drawing a recursion tree. The tree shows a lower bound with the following terms: n/2 ... n/10 ... n/50 ... etc. When I solve ...
0
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1answer
39 views

For what values of $\alpha$ does $1^{\alpha}$ does $1^{\alpha} = 1$. complex numbers

For what values of $\alpha$ does $1^{\alpha}$ does $1^{\alpha} = 1$. What are the possible values of $1^{\alpha}$? What are the values of $1^{\frac{1}{2}}$? (Hint: use the definition of ...
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votes
2answers
32 views

Rewrite the expression: $\log_2 x + 3 \log_2 y -5\log_2 z$ as a single logarithm: $\log_2 A$. Then the function A= [closed]

I can't seem to figure this problem out. I need to rewrite the expression $\log_2 x+3 \log_2 y-5 \log_2 z$ as a single logarithm $\log_2 A$ and its asking what the function $A $ will equal.
3
votes
3answers
68 views

Prove that $\frac{1}{\log_{2}{\pi}}+\frac{1}{\log_{\pi}{2}}> 2$

I have tried in many ways and i could not do it.
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3answers
63 views

Proof by induction: $\log n < n$ for $n ≥ 1$.

I was just wondering it is possible to prove this statement via mathematical induction? (I know you can do it via calculus but I want to specifically do it via induction). I have given it a go but am ...
2
votes
5answers
76 views

Limit as $x$ tend to zero of: $x/[\ln (x^2+2x+4) - \ln(x+4)]$

Without making use of LHôpital's Rule solve: $$\lim_{x\to 0} {x\over \ln (x^2+2x+4) - \ln(x+4)}$$ $ x^2+2x+4=0$ has no real roots which seems to be the gist of the issue. I have attempted several ...
1
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0answers
23 views

Why does this equality hold

let $P0,P1 \leq0$ $s.t$ $P0+P1=1$ (I am not sure if this assumption is required to prove the following equality. But for my application this holds). How does following hold true \begin{equation} ...
1
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1answer
35 views

Finding the equality of the natural logarithm to the limit and the infinite series (proof)

I'm trying to proof this equality which I found on this website: Euler-Mascheroni constant expression, further simplification $$\ln(n)=\lim_{M\rightarrow\infty}\sum\limits _{k=1}^{M}\sum\limits ...
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0answers
49 views

Give me the proof of this equality!? [duplicate]

I would like that someone can tell me stap for stap (mathematically proof) that this equality is true! I hope someone can help me and if you can I would be very thankfull ;) ...
2
votes
4answers
42 views

Why log(A) can be seen as %ΔA in economics?

We usually see deduce in economitrics changes $lnY$ to $\%\Delta Y$, say we have Cobb-Douglas function like this $Y=AK^αL^{1−α}$, in one of the book I read, it is changed to : But why can we do like ...
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1answer
15 views

Stuck on Double-Variabled Logarithm when solving this Sequences and Series question

I'm able to get to an inequality for the sum of the arithmetic sequence greater than the sum of the geometric sequence, and have solved the inequality by guess and check and have verified the ...
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1answer
77 views

A Logarithm Integral II [closed]

Does the integral \begin{align} \int_{0}^{1} (1-t)^{2} \, \ln^{k}(1-t) \, \ln^{m}(t) \, dt \end{align} have a compact form for $m = 1$, and $m=2$ ?
2
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0answers
110 views

Is this equal ? (I found it on this website)

I found this equation on this website! I would like to know it its true or not? And how can proof or disprove it?! Euler-Mascheroni constant expression, further simplification ...
1
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1answer
10 views

Find all solutions for a complex logarithm

$\log z = 6i$ I am working on a problem very similar. What I am seeing $\log z = \ln|z| + i(\theta + 2\pi n)$ for $n\in\mathbb{Z}$ What I am curious about, as if seen obvious to me that $ \log ...
0
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0answers
85 views

How can we proof that this is equal? About $ln(n)$

I found this on this website (Euler-Mascheroni constant expression, further simplification) without any explaining why this is equal can someone give me that? ...
3
votes
2answers
101 views

Is $g(x)=\log x$ convex function?

The graph of convex function is : In a book it is written that $g(x)=\log x$ is strictly convex function. So i searched for graph of $g(x)=\log x$ and found that Though it has been said that ...
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0answers
41 views

Analytically solving complicated integral involving logarithms.

I already asked a similar question a week ago and the comment I got helped me a lot with my progress, so that I now have new question to ask. I am stuck with solving a complicated integral and would ...
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2answers
46 views

Different results when integrating $1/(x \ln(x))$ partially/by substitution.

By substitution I get $ln(ln(x))$. Partially something completely different: $$\int \frac{1}{x \ln(x)} = \int \frac{1}{x} \frac{1}{\ln(x)} dx=\frac{\ln(x)}{\ln(x)} - \int -\frac{1}{x \ln(x) ^2} dx$$ ...
0
votes
1answer
53 views

A Trig Integral

Does the integral \begin{align} \int_{0}^{\pi/2} \cos(x) \, \ln\left( \frac{1 + a^{2} \sin(x)}{1 - a^{2} \sin(x)} \right) \, dx \end{align} have a closed form and what is changed if the limits are ...
1
vote
2answers
28 views

Prove the logarithmic inequality

Prove that: $(\log_{24}{48})^2+(\log_{12}{54})^2>4$ I tried to put $t=\log_23$ and get the equation $6t^4+32t^3+22t^2-84t-74>0$. But I can't do anything with it...
1
vote
1answer
49 views

Inequality with Logarithms!

I need some help solving this inequality for a question involving the number of bounces, $n$, of ball such that the max. height of the ball is less than 5cm. This is the equation I have gathered from ...