Questions related to real and complex logarithms.

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5
votes
1answer
98 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
232 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
54 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
266 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
137 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
40 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
302 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
72 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: $${\log_2n \over n} >{ 1 \over 8}$$
8
votes
1answer
799 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
43 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
476 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
46 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, ...
2
votes
1answer
44 views

Making logarithmic function go higher

I am looking at logarithmic functions, and, lets say, log2 (x+3) is having a bit of a growth rate between 0-10 values of ...
2
votes
1answer
112 views

Online logarithm drawing

I am looking for a site that will give me the output of my logarithms. What I want to do, is I want to input, in example log(2), and I want it to draw an output ...
0
votes
1answer
41 views

Algebra involving powers of two when solving for cardinality involving power sets

What's the proper way of solving this problem (self study)? Find $m$ if $|A|=m$, $|B|=m+2$ and $|\mathcal{P}(B)| - |\mathcal{P}(A)| = 48$ Since the numbers are small, I can get $m$=4 pretty much by ...
1
vote
2answers
36 views

Quotient of Arguments to a Logarithm

The problem I'm working on is: There are $p, q > 0$ satisfying $\log_9p=\log_{12}q=\log_{16}(p+q)$. Find $q/p$. I set all of the logarithms equal to a variable $t$, so I could say ...
6
votes
2answers
436 views

How do I solve for $x$ in $\ln(x)\ln(x) = 2 +\ln(x)$

How do I solve for $x$? $$\ln(x)\ln(x) = 2 +\ln(x)$$
2
votes
5answers
580 views

Prove $e^{\ln{x}} = x$

Is it possible to prove $e^{\ln{x}} = x$ for a student or can you only say that exponentiation is defined to be the inverse of natural logarithm and stop there?
1
vote
1answer
75 views

domain of function with logarithmic terms.

what will be the domain of function given below? $$y=1+3(\log(\sin(x))+\log(\csc(x)))$$ in book it is given this is valid for the values of angles of 1st and 2nd quadrant only. why this function is ...
4
votes
3answers
620 views

How to solve $e^x = 2$

I know that $\ln(x)$ is the inverse of the exponential function $a^x$. So I thought that $$ e^x=2 \Leftrightarrow x = \ln(2) $$ but my calculator says $x = \ln(2) + 2 i \pi n$, where $N \in ...
1
vote
1answer
88 views

Solving $\log_x(x^2+4x)>-1$

I'm stuck looking for a solution for this. Any hint? $$ \log_x(x^2+4x) > -1 $$ It looks like $$ x^2+4x > 1/x $$ which I cannot solve.
1
vote
3answers
136 views

Why can a Complex Logarithm have infinitely many values?

What does this mean, that "Due to the periodicity of the trigonometric functions, a complex logarithm can have infinitely many values"? $\ln z=\ln (\cos x +i\sin x)=?$
0
votes
1answer
66 views

What is the sum of this series?

The series is as follows: $$\lg n + lg (n - 2) + lg (n - 4) + \ldots + lg (2)$$ Thanks.
3
votes
2answers
213 views

Alternatives to percentage for measuring relative difference?

If the value of something changes from $\;a\;$ to $\;b\;$, their relative difference can be expressed as a percentage: $$ \newcommand{\upto}{\mathop\nearrow} (D0) \;\;\; a \upto b \;=\; (b-a)/a ...
2
votes
2answers
83 views

Inequality $C\lceil\log{n}\rceil! \geq n^k$

I've been struggling to prove there exist $C$ for $n, n_{0}, \forall k >0 \in \mathbb{R}$ such that $\forall n > n_{0}$: \begin{equation}C\lceil\log{n}\rceil! \geq n^k\end{equation} As you ...
1
vote
3answers
123 views

trying to solve $2^x \equiv 9 \pmod{13}$

I'm trying to solve $$ 2^x \equiv 9 \pmod{13} $$ so I tried to define all numbers for $x$ which match this requirement and I came up with this equation: $$ \sqrt{\sin(((x)-13/2-9)*\pi/13)^2} $$ ...
4
votes
1answer
80 views

A double sum and its relation to a simple sum, is this an identity for any complex number $S=a+i b$ and any integers n and t?

Does: $$\sum _{m=1}^t \lim_{s\to \text{S}} \, \zeta (s) \sum _{k=(m-1) n+1}^{m n} \frac{1-\text{If}[k \bmod n=0,n,0]}{k^{s-1}}$$ equal: $$\lim_{s\to \text{S}} \, \zeta (s) \sum _{k=1}^{n t} ...
2
votes
2answers
267 views

Show that $n>(\log n)^k$ for sufficiently large values of $n$ [duplicate]

How do you show that $n>(\log n)^k$, $k \in Z_+$ for sufficiently large values of $n$? This is a part of a larger problem that I want to solve, so I would be thankful if someone could show me the ...
1
vote
1answer
53 views

$100(1.2)^t=\alpha t$. How do I solve for t?

I'm playing with a population model, and I wondered how to find the $t$ when the population goes to zero, with regard to $\alpha$. I've ended up with this, and I don't know how to solve for t. Can ...
0
votes
3answers
155 views

If $\ln a+\ln b=\ln c$, is $a+b=c$?

I'm going to rephrase this because I seem to be confusing people. If I have $a+b=c$ I can say $\ln a+\ln b=\ln c$ But if I have $\ln a+\ln b=\ln c$ I can't say $a+b=c$ Why?
4
votes
4answers
145 views

$\ln(-1) - \ln(-2)$ is it definable or have answer?

As the title says I type in google and the number say -0.693... Is it equal to ln(1/2)? Am I misconcept anything?
1
vote
2answers
49 views

Two possible answers for x

I was trying to solve a question on maxima-minima and I finally ended up getting this equation: $$\ln(1/x)=1;$$ If I take anti-log on both sides I get 1/x=e and therefore x=1/e. But if ...
1
vote
2answers
128 views

Deriving logarithmic identities

Wikipedia says: The numerical value for logarithm to the base $10$ can be calculated with the following identity: $$\mathrm{log_{10}}(x) = \frac{\mathrm{ln}(x)}{\mathrm{ln}(10)} \\ ...
1
vote
2answers
136 views

$e^{\ln(-2)} = -2$ but $\ln(-2) = \ln 2+i\pi$. How does this work?

I'm messing with exponential growth functions. I noticed that I can write $y(t)=y(0)\alpha^t$ as $y(t)=y(0)e^{\ln(\alpha)t}$ (and then I can go ahead and replace $\ln(\alpha)$ with $\lambda$.) ...
0
votes
2answers
44 views

Converting to logarithm

What logarithm rule can convert: $$\left(\frac n4\right)^i = 1$$ to: $$i = log_4(n)$$ When I view cheat-sheets for logarithm rules, I only see conversions where both sides of the equation have log ...
0
votes
1answer
34 views

Growth rate of $n^2$ vs $(\log_3(n))^3$

Which grows faster, $n^2$ or $(\log_3(n))^3$? How do I figure out which grows faster in general in these kinds of situations?
2
votes
1answer
92 views

Questions about $\ln(z)$ recurrence and fixed points.

Define property $A_R$ for an analytic function $f(z)$ as $1)$ $f(z)=0$ has exactly one solution being $z=0$ for $|z|<R$ where $R$ is a radius. And $f(z)$ is analytic within the radius $R$ ...
2
votes
1answer
46 views

Question about fixpoints and zero's on the complex plane.

Define property $A$ for an entire function $f(z)$ as $1)$ $f(z)=0$ has exactly one solution being $z=0$ $2)$ $f(z)=z$ has exactly one solution $=>z=0$ (follows from $1)$ ) $3)$ $f(z)$ is not a ...
-1
votes
1answer
40 views

Stuck in a Exercises when lerning for a test [closed]

I am Preparing for an Exam, and this is one of my Exercises that I tried to solve, I know that the answer is : $x = 2,-1$ but Every time I try to solve, my answer is Different from the Right answer. ...
3
votes
3answers
142 views

Simplifying Logarithms

I am Preparing for an Exam, and this is one of my Exercises that I tried to solve, I know that the answer is : $-2<x\le1\frac{1}{2}$. but Every time I try to solve, my answer is Different from ...
1
vote
1answer
69 views

How to solve the log Equation

I tried to solve the Equation but unfortunately Whatever I try, I can't solve the Equation $$100^{\log x}=3x^3$$