Questions related to real and complex logarithms.

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187 views

Homework with logarithms

I'm stuck on continuing the next exercise: Considering: $\log_{c}a = 3$ $\log_{c}b = 4$ and: $$ y = \frac{a^{3}\sqrt{b \cdot c^{2}}}{2} $$ What's the value of $\log_{c}y$ ...
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2answers
135 views

Is $2^{\log_2(-5)}$ defined?

As far as I know for $\log_2 x$ to be defined $x$ must be higher than 0. However when I enter $2^{\log_2(-5)}$ into wolframalpha it gives result $-5$. Is it mistake?
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1answer
88 views

A logarithmic equation?

$$\mathcal{O} \left(3^{\log_2(n)} \right) = \mathcal{O} \left(n^{\log_2(3)} \right)$$ Does anyone have any idea how the right side was arrived at? (The $\mathcal{O}$ is Big-$\mathcal{O}$ notation)
4
votes
1answer
192 views

Name for logarithm variation that works on non-positive values?

I've come up with the following variation of a logarithm, intended to work on values that can be 0, or can grow exponentially from zero in either positive or negative direction. $$myLog(x) = ...
4
votes
3answers
147 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 ...
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4answers
144 views

Prove $\ln{(\frac {x}{y})} = \ln{x} - \ln{y}$ using the definition $\int_1^x\frac{1}{t}dt = \ln{x}$.

Prove $\ln (\frac{x}{y}) = \ln{x} - \ln{y}$ using the definition $\int_1^x\frac{1}{t}dt = \ln{x}$. I am able to prove $\ln{xy} = \ln{x} + \ln{y}$, and $\ln{x^r} = r\ln{x}$, but with this one, I am ...
4
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4answers
146 views

How to find the limit of $\dfrac{\ln(\ln(\frac{n}{n-1}))}{\ln(n)}$?

How do you find $$\lim_{n \to\infty} \dfrac{\ln(\ln(\frac{n}{n-1}))}{\ln(n)}$$ I know it's $-1$, but I had to plot it.
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2answers
144 views

Derivative of ${ x }^{ x }$ without logarithmic differentiation

With logarithmic differentiation, it is quite simple to compute the derivative of $x^x$: $$y=x^x$$ $$\ln {y} =x \ln{x}$$ $$\frac {1}{y} \frac {dy}{dx} = \ln{x} +1$$ $$\frac {dy}{dx} ...
4
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3answers
169 views

Summation of a series.

I encountered this problem in Physics before i knew about a thing called Taylor Polynomials My problem was that i had to sum this series : ...
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3answers
93 views

Does $\operatorname{Log}(1+i)^2 =2\operatorname{Log}(1+i)$

And similarly, does $\operatorname{Log}(1-i)^2=2\operatorname{Log}(1-i)$? If we were dealing with real numbers, it would hold. But I'm guessing that the fact that there are imaginary numbers involved ...
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2answers
124 views

Limiting value of $\lim \frac{1}{k}\sum_{n=1}^k \frac{p(n+1)-p(n)}{\log p(n)}$

Empirically it seems $$\lim_{k\to \infty} \frac{1}{k}\sum_{n=1}^k \frac{g(n)}{\log p(n)} = 1\tag{1} $$ in which p(n) is the nth prime and g(n) is the prime gap $p(n+1)-p(n).$ Cramer conjectured ...
4
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2answers
240 views

Natural logarithm, equivalent function

I'm developing a software with a tool unable to "recognize" the ln(), so is there a way to get the equivalen to ln() using someones of functions below? • sin1(a) • cos1(a) • tan1(a) • ...
4
votes
3answers
173 views

How to prove this ln inequality?

I have the following inequality, which (supposedly) holds for every $x\in\mathbb{R}$: $$ 1+x\ln\left(x+\sqrt{1+x^{2}}\right)\geq\sqrt{1+x^{2}} $$ I've been struggling to find some known inequalities ...
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2answers
626 views

What does $\log^{2}{x}$ mean?

What is it used for and why doesn't it equal $\large\log{x^2}$?
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1answer
90 views

Does the logistic function really uniquely satisfy this?

It is said that the logistic function (denoted $y(u)$ below) is derived from the relation: $$\frac{dy}{du}=y(u)(1-y(u))$$ Does $y(u)=\frac{1}{1+e^{-u}}$ really uniquely satisfy this? I don't see ...
4
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1answer
401 views

Exponential and log functions compose to identity

How to prove that the exponential function and the logarithm function are the inverses of each other? I want it the following way. We must use the definition as power series, and must verify that all ...
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4answers
113 views

Simple looking log problem

How would I solve this for $x$? The original problem is $$x+x^{\log_{2}3}=x^{\log_{2}5}$$ I have tried to reduce it down to this, $$x^{\log_{10}3}+x^{\log_{10}2}=x^{\log_{10}5}$$ I have been ...
4
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2answers
70 views

How to Solve for Zero

$$4x^2e^{-x^2}-2e^{-x^2}=0$$ I took out a common factor of $2e^{-x^2}$ which got me to: $2e^{-x^2}(2x^2-1)=0$ I'm not sure if taking out the common factor helped at all and I don't know where to go ...
4
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3answers
180 views

Solve $3\log_{10}(x-15) = \left(\frac{1}{4}\right)^x$

$$3\log_{10}(x-15) = \left(\frac{1}{4}\right)^x$$ I am completely lost on how to proceed. Could someone explain how to find any real solution to the above equation?
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4answers
1k views

How do I cube/square a logarithm?

Btw, please don't give me the answer. I just wanna know how to raise a logarithm to its cube cause I'm stuck in this part, but don't solve it for me. $$\log \sqrt[3]x = \sqrt[3]{\log x}$$ I tried ...
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1answer
1k views

$\log_7 n$ is either an integer or an irrational number

Show that $\log_7 n$ is either an integer or an irrational number where n is a positive number. I assumed that it is rational and tried to get a contradiction for $\log_7 n = a/b$, where b does ...
4
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2answers
243 views

How can I solve $x^x = 5$ for $x$? [duplicate]

Possible Duplicate: Is $x^x=y$ solvable for $x$? I've been playing with this equation for a while now and can't figure out how to isolate $x$. I've gotten to $x \ln x = \ln 5$, which seems ...
4
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5answers
69 views

Calculate ln(x) using 8-digit calculator

I have a bit of a unique problem. Well, maybe not a problem because I'm really just curious about it, but... I have a simple 8 digit calculator. It has +, -, x, /, and a constant operation function. ...
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1answer
46 views

Is $n^{\log c} = c^{\log n}$ true?

Is $n^{\log c}$ the same as $c^{\log n}$? If so, please explain.
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3answers
230 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 ...
4
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1answer
489 views

If $ax^2-bx+c=0$ has two distinct real roots lying in the interval $(0,1)$ $a,b,c$ belongs to natural prove that $\log_5 {abc}\geq2$

If $ax^2-bx+c=0$ has two distinct real roots lying in the interval $(0,1)$ with $a, b, c\in \mathbb N$, prove that $\log_5 {abc}\geq2$. The equations I could form are: 1) $f(0)>0$ and ...
4
votes
1answer
152 views

How do you solve this equation: $10 = 2^x + x$?

Is it possible to solve this equation? \begin{align} a &= b^x + x \\ a-x &= b^x \\ \log_b(a-x) &= x \end{align} If $a$ and $b$ are known, how do you find $x$?
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1answer
51 views

Looking for help in understanding Jitsuro Nagura's analysis of the upper bound for $\psi(x)$

I'm working on understanding Nagura's analysis of the upper bound for $\psi(x)$ which is done in Lemma 2. I am unclear on one step of his reasoning. With Lemma 1, he establishes for $x \ge 2000$: ...
4
votes
2answers
100 views

How to isolate x in this equation?

Suppose we have this equation: $y = 1 + x + \lceil \log_2(x)\rceil$ where $x$ is an integer > $0$. How can we get $x$ as a function of $y$ (basically isolate $x$)? I don't understand how to handle ...
4
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2answers
178 views

I am trying to solve the inequality $\log_{\log{\sqrt{9-x^2}}} x^2 <0$

I am trying to solve the inequality $$\log_{\log{\sqrt{9-x^2}}} x^2 <0.$$ I got $\mathrm{S.S}=(-\sqrt8 ,-1)\cup( 1,\sqrt8)$, but a friend got $\mathrm{S.S}=(-1,1)- \{0\}$. Please, what is ...
4
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3answers
208 views

Proving convergence of $\int^{\infty}_{0}\frac{\ln{x}}{1+x^{2}}\,dx$

I was trying to prove the convergence of the following integral: $$\int^{\infty}_{0}\frac{\ln{x}}{1+x^{2}}\,\mathrm{d}x$$ The only way (and indeed quite a convenient one) that came to mind was using ...
4
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4answers
259 views

How to find the limit $\lim \limits_ {x \to+\infty} \left [ \frac{4 \ln(x+1)}{x}\right]$

Solve $\space \begin{align*} \lim_ {x \to+\infty} \left [ \frac{4 \ln(x+1)}{x}\right] \end{align*}$. I did this way: $$\begin{align*} \lim_ {x \to+\infty} \left [ \frac{4 \ln(x+1)}{x}\right] ...
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1answer
1k views

How does Knuth's algorithm for calculating logarithm work?

I had a look at Knuth's The Art of Computer Programming, book 1. In chapter 1, section 1.2.2, exercise 25, he presents the following algorithm for calculating logarithm: given $x\in[1,2)$, do the ...
4
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1answer
403 views

What constants do I need to create this specific logarithmic spiral?

please bear with me as I'm not a mathematician and this is difficult to word properly. :] I need the equation for a logarithmic spiral (let's call it $S(\theta)$) that meets certain constraints for a ...
4
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5answers
168 views

How do I solve such logarithm

I understand that $\log_b n = x \iff b^x = n$ But all examples I see is with values that I naturally know how to calculate (like $2^x = 8, x=3$) What if I don't? For example, how do I solve for $x$ ...
4
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2answers
64 views

Given $1<a<b<c$ prove $\log_a\log_ab+\log_b\log_bc+\log_c\log_ca>0.$

Given $1<a<b<c$ prove $$ \log_a\log_ab+\log_b\log_bc+\log_c\log_ca>0. $$ How to approach problems like this? I tried usual transformations but no help. I guess I have to use ...
4
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2answers
276 views

How to calculate number of digits of a large number?

Does anyone know any efficient ways of finding the number of digits in the large number $N = 4^{4^{4^4}}$? Thanks.
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3answers
103 views

Best approximation for $\displaystyle \sum_{k=2}^n\ln\ln k$?

I need the best approximation for $\displaystyle{\sum_{k = 2}^{n}\ln\left(\ln\left(k\right)\right)}$. Any suggestion or hint is welcomed. I derived $n\ln\left(\ln\left(n!\right) \over n\right)$ so ...
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1answer
72 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 ...
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4answers
122 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?
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1answer
176 views

Solving a transcendental equation consisting of a quadratic part and a part involving inverse Lambert W functions

Question statement I would like to solve the following equation in the two variables $x$ and $y$: \begin{gather} 0 = x^2 - a y^2 + i b [x y - W^{-1}(x)W^{-1}(y)] , \end{gather} where $a$ and $b$ are ...
4
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3answers
128 views

Write the expressoin in terms of $\log x$ and $\log y \log(\frac{x^3}{10y})$

What is the answer for this? Write the expression in terms of $\log x$ and $\log y$ $$\log\left(\dfrac{x^3}{10y}\right)$$ This is what I got out of the equation so far. the alternate form assuming ...
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1answer
100 views

Trouble proving ln inequality

I looking for help with proving the following inequality. Any relevant logarithmic identities would be great. Tried differentiating and taking limits and I'm lost as to how to approach this. $$\frac ...
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2answers
325 views

Solve logarithmic equation

I'm getting stuck trying to solve this logarithmic equation: $$ \log( \sqrt{4-x} ) - \log( \sqrt{x+3} ) = \log(x) $$ I understand that the first and second terms can be combined & the logarithms ...
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1answer
108 views

If $ \log \log x =1$, then is it true that $e^e =x$?

I worked out this question, and I wanted to see if my understanding of the concepts involved is sound. Solve for $x$ $$\ln(\ln(x))=1$$ $$e^1=\ln(x)$$ $$e^e=x$$ Since any number raised to $1$ is ...
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2answers
252 views

Can all logarithm problems be solved algebraically?

Trying to solve $\log_2(x-1)=\log_3(x+1)$ and can't seem to get it algebraically. Tried changing bases, moving things around, but can't seem to crack it.
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1answer
296 views

The definition of the logarithm.

One usually gets several definitions of the logarithm along his studies. You might be first introduced to the exponential and then told that the logarithm is its inverse. You might be given $$\log ...
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1answer
354 views

Help understanding this formula on mutual information (used in bioinformatics)

I'm a bit lost on understanding this formula in my bioinformatics text, and I appreciate any tips or advice. Mutual Information, $\operatorname{MI}(X; Y)$ is: $$ \mu = \sum_x \sum_y p(xy) ...
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1answer
154 views

Show the correctness of a logarithmic inequality

Let $p_1>p_2$ and $n_1>n_2$ be positive numbers. I want to show that, $$ \frac{\log \left(\frac{p_1}{n_1}+1\right)}{\log \left(\frac{p_2}{n_2}+1\right)}\leq \frac{\log ...
4
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3answers
820 views

Find a branch of $f(z)= \log(z^3-2)$ that is analytic at $z=0$.

Find a branch of $f(z)=\log(z^3-2)$ that is analytic at $z=0$. Can anyone help me on this question? I have no idea how to find a branch. The definition of branch given in lecture is $F$ is a branch ...