Questions related to real and complex logarithms.

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0
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2answers
127 views

Notation of logarithm and its exponent

I am little confused about this notation, $\log^3 n$. Does it mean $(\log n)^3$ or $\log (\log (\log n))$?
0
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2answers
66 views

Which is bigger $x^{log(x)}$ OR $(log(x))^x$

I'm trying to find out which is bigger $x^{log(x)}$ OR $(log(x))^x$ As $x \to \infty$ I tried to take the log of both but I didnt't reach any where.
0
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2answers
428 views

How do I prove that the limit as one function goes to infinity is equal to another function?

I was playing around with the integral $\int x^ndx$ and noticed that it is always $\frac{x^{n-1}}{n-1}+c$ except for the singular case $n=-1$. So I could pick $n$ arbitrarily close to $-1$, and the ...
1
vote
1answer
44 views

Deriviative of natural log help finding

$$y=7\ln\frac{11}x$$ I need to use the product rule please $$\frac{d}{dx} (7) \cdot(\ln(11/x) + \frac{d}{dx}\left(\ln\frac{11}x\right) \cdot 7$$ then $$0+\frac{d}{dx}\frac{11}x \cdot7$$ what do ...
1
vote
2answers
68 views

Solving this logarithm equation?

How do I solve this equation using common logarithms? $\log x = 1-\log(x-3)$
0
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1answer
74 views

From $ \sum^\infty_{\lfloor \log n \rfloor + 1}n/{2^r} $ to $ \sum^\infty_{r=0}1/2^r $?

$$ E[h] = E[\sum^\infty_{r=1}I_r] = \sum^\infty_{r=1}E[I_r] $$ $$ = \sum^{ \lfloor \log n \rfloor}_{r=1}E[I_r] + \sum^\infty_{\lfloor \log n \rfloor + 1}E[I_r] $$ $$ \leq \sum^{ \lfloor \log n ...
1
vote
0answers
113 views

Matrix logarithm of sum

Let $D$ be a diagonal, $n\times n$ complex matrix, and let $X$ be an $n\times n$ complex matrix. Is there is simple formula for \begin{align} \log(D + \epsilon X) \end{align} as a power series in ...
4
votes
3answers
256 views

$\log(n)$ is what power of $n$?

Sorry about asking such an elementary question, but I have been wondering about this exact definition for a while. What power of $n$ is $\log(n)$. I know that it is $n^\epsilon$ for a very small ...
0
votes
1answer
84 views

Solving logarithms with different bases?

How would I go about getting an exact value for a question like: $\log_8 4$ I know that $8^{2/3} = 4$ but how would I get that from the question?
0
votes
5answers
116 views

How do I reduce this: $\frac{2}{x\ln(4)}\quad ?$

$$\text{Given}\;\;\frac{2}{x\ln(4)}, \;\;\text{how does it reduce to}\; \frac{1}{x\ln(2)}\quad ?$$
3
votes
2answers
193 views

Prove that $\log^25 + \log^27 > \log12$.

Prove that $\log^25 + \log^27 > \log12$. What I tried so far: $\log^25 + \log^27 > \log3 + \log4$ $(\log5 + \log7)^2 - 2 \cdot \log5 \cdot\log7 > \log3 + \log4$ But it seems that I'm not ...
37
votes
1answer
821 views

Integral $\int_0^1\frac{x^9\left(x^4+x^2-x-1-5\ln x\right)}{\left(x^{10}-1\right)\ln x}\mathrm dx$

A friend of mine sent me an integral that she had not been able to crack, and me neither. It comes with a result, but without a proof (I suppose it originated in some math contest). Could you please ...
0
votes
1answer
31 views

For each of series find the smallest $k$, that $a_n = O(n^k)$

Hey I need you to check my solutions: a) $a_n = (2n^{81.2}+3n^{45.1})/(4n^{23.3}+5n^{11.3})$ This one is done from $\sum_{i=1}^{k} O(a_i(n)) = O(max\lbrace a_i,..,a_k \rbrace )$ So it's ...
0
votes
2answers
85 views

Finding an equation for a growth formula

Given a tree that has three nodes each level I want to find the formula that predicts the number of all nodes with a given tree height. I fitted the data into Numbers with an exponential function ...
2
votes
2answers
99 views

Evaluating $\int _{-1}^{e} \frac{1}{x}dx$

Here very easily by the Fundamental Theorem of Calculus $$\int _{-1}^{e} \frac{1}{x}dx=\ln(e)-\ln(-1)$$ From Euler's identity $e^{i \pi}$=-1 we can easily deduce that $\ln(-1)=i \pi$. Thus the ...
1
vote
2answers
444 views

absolute value of logarithm

I have a problem with understand how function $2^{|\log_{1/2}x|}$ obtains values for the negative $x$ ? I thought that there is the assumption that $x>0$ but wolframalpha shows chart that for ...
12
votes
2answers
337 views

Conjecture $\int_0^1\frac{\ln\left(\ln^2x+\arccos^2x\right)}{\sqrt{1-x^2}}dx\stackrel?=\pi\,\ln\ln2$

$$\int_0^1\frac{\ln\left(\ln^2x+\arccos^2x\right)}{\sqrt{1-x^2}}dx\stackrel?=\pi\,\ln\ln2$$ Is it possible to prove this?
27
votes
2answers
960 views

Are there other cases similar to Herglotz's integral $\int_0^1\frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\ \mathrm dt$?

This post of Boris Bukh mentions amazing Gustav Herglotz's integral $$\int_0^1\frac{\ln\left(1+t^{\,4\,+\,\sqrt{\vphantom{\large A}\,15\,}\,}\right)}{1+t}\ \mathrm ...
0
votes
2answers
170 views

Derivative of f(x)=arth(lnx)

I'm struggling with finding derivative of f(x)=arth(lnx). I've done following: x=th(y) ...
1
vote
1answer
179 views

Logarithm Problem : Find the number of real solutions of the equation $2\log_2\log_2x+\log_{\frac{1}{2}}\log_2(2\sqrt{2}x)=1$

Find the number of real solutions of the equation $2\log_2\log_2x+\log_{\frac{1}{2}}\log_2(2\sqrt{2}x)=1$ My approach : Solution : Here right hand side is constant term so convert it into log ...
0
votes
3answers
923 views

Something to the power of a logarithm

This is probably a very obvious question, but here goes... An answer in my textbook claims that $$3^{\log n} = n^{\log 3}$$ and that $$4n^2 (3/4)^{\log n} = 4n^{\log 3}$$ Why, using more basic ...
8
votes
1answer
237 views

Integral $\int_0^\infty\frac{dx}{\frac{x^4-1}{x\cos(\pi\ln x)+1}+2x^2+2}$

I need your help with this integral: $$\int_0^\infty\frac{dx}{\frac{x^4-1}{x\cos(\pi\ln x)+1}+2\,x^2+2}.$$ I wasn't able to evaluate it in a closed form, although an approximate numerical evaluation ...
0
votes
2answers
47 views

Logarithmic functions?

I am stuck on this question which as follows: $\log(x) + \log(x-3) = \log(10x)$ I have tried the following and not sure if I am doing it correctly... 1) $$ \log(x) + \log(x-3) = \log(10x) ...
0
votes
3answers
41 views

Using Logarithms

\begin{align*} -2^{n-1} \ln2 &= -100 \ln 10\\ &\\ -100 \ln 10 &= -230\\ &\\ \dfrac{-230}{\ln (2)} &= -333\\ &\\ -2^{n-1} &> -333\\ &\\ (n-1) \ln(-2) &> ...
0
votes
2answers
55 views

Adding Logarithms

Studying for my midterm. Solve the following algebraically: $$\log_2x+\log_2(x+4)=5$$ So I know that $\log_b(mn)=\log_b(n)+\log_b(m)$ therefore: $$5=\log_2(x(x+4))$$ $$\text{or}$$ ...
0
votes
4answers
79 views

Solving for x in a equation involving natural logarithms

How would I solve for x in this equation here: $$\ln(x)+\ln(1/x+1)=3$$ I realize that the answer is $e^3-1$, but I am not sure as to how to get it. Any input is appreciated.
0
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2answers
72 views

How does my professor go from this logarithm to the next?

In the above picture, how does he go from the third-last line to the second last?
0
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2answers
162 views

How to solve log decimal

How do I solve? $$ x = \log_{10} 5$$ Until I understood until for now, it's same as: $$ 10^x = 5 $$ and $x$ will be a value $> 0$ and $< 1$ because if it's $1$, the value $= 10$ But someone ...
1
vote
1answer
95 views

How to solve $2^x = 36$

I need to solve $\log$ of $36$ in base $2$ The logarithm result $= x$. $$ \log_ 2 36 = x. $$ How do I determine value of $x$ in $$ 2^x=36 $$ I don't know how do it, since there's perfect square of ...
5
votes
1answer
104 views

Maximize $W(x) - (\ln(x) - \ln{\ln{x}})$

How can you maximize $f(x) = W(x) - (\ln(x) - \ln{\ln{x}})$ for $x\geq 2$? Numerically the answer seems to be at around $x \approx 41$ where you get $f(41) \approx 0.31$. Mathematica suggests the ...
0
votes
1answer
27 views

Why does $\exp\left[W\left(b\left(\ln{n}\right)^2\right) - \ln{b} - \ln{\ln{n}}\right] = \frac{\ln{n}}{W(b\ln^2{n})}$?

Why does $$\exp\left[W\left(b\left(\ln{n}\right)^2\right) \; - \; \ln{b} - \ln{\ln{n}}\right] = \frac{\ln{n}}{W(b\ln^2{n})}\;?$$ $W$ is the Lambert-W function and all variables are real and positive. ...
0
votes
1answer
86 views

How to solve $q= \frac{\ln{n}}{\ln{b} + \ln{q}+\ln\ln{n}}$

I have come across this equation recently. All the variables are positive and real too. $$q= \frac{\ln{n}}{\ln{b} + \ln{q}+\ln\ln{n}}.$$ Under what conditions can this be solved for $q$?
6
votes
6answers
235 views

Inequality, what is wrong with $\log(-1) = - \log(-1)$?

Can anyone tell me what is wrong with the following line of argument: $$ \log(-1) = \log(-1) - \log(1) = - \bigg( \log(1) - \log(-1) \bigg) = - \log \Big( \frac{1}{-1} \Big) = - \log(-1) $$ ...
1
vote
2answers
56 views

Logarithm problem : Prove that $log_{3^2} \frac{1}{2} > 0$

Logarithm problem : Prove that $log_{3^2} \frac{1}{2} > 0$ My approach : $log_{3^2} \frac{1}{2} > 0$ $\Rightarrow \frac{1}{2} log_3 \frac{1}{2} >0$ $\Rightarrow \frac{1}{2} [ log_3 1 ...
1
vote
1answer
79 views

How does $7\log(8x) = 7\ln8x$?

I was working on some math homework with a program called scientific notebook. I was check that I was writing something correctly. The original equation is $(\log(x^4)+\log(x^5))/\log(8x)=7$ I then ...
1
vote
1answer
57 views

What am I doing incorrectly; logarithms?

We have an increasing number of books on a bookshelf. Every year, 2 books are added and each book is twice as long as the previous book. At the beginning of 1935 the volume was 1 cm thick. We define ...
7
votes
1answer
1k views

where do exponential and logarithmic functions intersect?

If $0<a<1$, then the graphs of $y=a^x$ and $y=\log_a(x)$ intersect at some point $(t(a),t(a))$. Does this function $t(a)$ have any nice expression? How much do we know about this function, ...
2
votes
2answers
72 views

Integral $\int_{-1}^{0}\dfrac{1}{x(x^2+1)}$

Suppose I have to compute $\int_{-1}^{0}\dfrac{1}{x(x^2+1)}$. I use partial fractions to get $\int_{-1}^{0}\left(\dfrac1x-\dfrac{x}{x^2+1}\right)$, which integrates to $\log(x)-\log(x^2+1)$. Now, the ...
1
vote
1answer
84 views

base transformation rule significance in finding big o notation

Recall the equivalence: $$m=b^k \implies k = log_bm$$ as well as the base transformation rule: $$log_am=(log_ab)(log_bm)$$ Are the following true or false? (a) $log_2n$ is $O(log_3n)$ (b) ...
2
votes
1answer
69 views

Solve for $n$ in $(4/3) ^n = 6439 / 3072 $

It is a series and sequences question. I worked it up to here but I'm stuck at this point. $$\left(\dfrac{4}{3}\right)^n = \dfrac{6439}{3072}$$ Solve for $n$. Thanks
18
votes
2answers
269 views

A closed form for $\int_0^1\frac{\ln(-\ln x)\ \operatorname{li}^2x}{x}dx$

Let $\operatorname{li}x$ denote the logarithmic integral $^{[1]}$$^{[2]}$$^{[3]}$: $$\operatorname{li}x=\int_0^x\frac{dt}{\ln t}$$ and $$I=\int_0^1\frac{\ln(-\ln x)\ ...
1
vote
1answer
141 views

Understanding a question on iterated logarithms

I have in front of me a math problem that I do not understand. That's to say, I don't understand what is being asked of me. Problem: We can define $\log_2**(x) = log_2*(log_2*(x))$ and the function ...
0
votes
1answer
42 views

Basic demonstration concerning the natural logarithm…

Recently saw a question that asked me to show that In(√2 - 1) = - In (√2 + 1). How can I demonstrate that the LHS equals the RHS?
3
votes
1answer
67 views

Next number that is a 1 and zeroes

There's probably a much better way of expressing this, but I don't know it, so I guess that's part of the question too. I'm not even sure what to tag it. How do I find the next number greater than a ...
-1
votes
1answer
312 views

Logarithms and big O notation

Recall the equivalence: $$m=2^k \implies k=log_2m$$ (a) Consider the sequence: $a_1=1, a_{k+1}=2a_k$ what is the smallest k for which $a_k \geq n$? Your answer should be a function of n, and you can ...
0
votes
3answers
76 views

Solving inequality having log

I am struggling to solve this inequality involving logarithm. How to find out values of $n$ for which below inequality holds good: $${\log_2n \over n} >{ 1 \over 8}$$
9
votes
1answer
967 views

Why is the Logarithm of a negative number undefined?

The Definition of a Logarithm is: If $$x^y=a$$ Then $$\log_xa=y$$ Given this definition, since $$e^{i\pi}=-1$$ Then shouldn't $\ln(-1)=i\pi$? What is wrong with it?
0
votes
1answer
46 views

Exponential growth/reduction of Laser Intensity question?

Hi i have a pretty simple question but I am not quite sure on how to solve/approach it. THe question: "The Intensity of a laserbeam declines with the penetrationdepth into matter exponentially. At ...
26
votes
1answer
480 views

Integral $\int_0^1\ln\ln\,_3F_2\left(\frac{1}{4},\frac{1}{2},\frac{3}{4};\frac{2}{3},\frac{4}{3};x\right)\,dx$

I encountered this scary integral $$\int_0^1\ln\ln\,_3F_2\left(\frac{1}{4},\frac{1}{2},\frac{3}{4};\frac{2}{3},\frac{4}{3};x\right)\,dx$$ where $_3F_2$ is a generalized hypergeometric function ...
2
votes
1answer
48 views

How to decide for the number of decimals for rounding

I am computing some logarithms and have a problem for the result's presentation. These are my numbers. 1.0986122886681098, 1.0986122886681098, 0.0, 0.8109302162163288, 1.0986122886681098, ...