Questions related to real and complex logarithms.

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1answer
80 views

Two $\psi$ functions

This is either a notation/history question or a point of confusion. In (for example) Ramanujan's proof of Bertrand's postulate, he uses the following notation: $\log [x]!$ means $\log ([x]!),$ in ...
0
votes
2answers
41 views

How to evaluate this formula

How can I evaluate $1/e^{\ln(x)}$? I really don't have experience on this and appreciate if you can explain it to me. Thanks.
4
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2answers
311 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) • ...
1
vote
1answer
248 views

Logarithm in an inequality: is it solvable?

Can anyone help me understand what happens to the following inequality once I apply a logarithm to all three parts? $$ - \varepsilon < 2^{\frac{1}{x}} < \varepsilon \longrightarrow \ln(- ...
1
vote
1answer
60 views

Have I made a mistake in manipulating this equation?

I found this manipulation of the equation for the error bound of the Bisection method here - http://www.maths.uniswa.sz/docs/m311/bisection.pdf The error bound is $$\frac{b-a}{2^{n+1}} < ...
2
votes
1answer
100 views

Complex logarithm and injectivity

Please forgive the trivial nature of this question: let U be a connected domain inside the punctured unit disk so that every curve inside it has winding number zero around the origin. Is the complex ...
1
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1answer
84 views

Evaluate $\sum_{n=1}^{1024}\left \lfloor \log_2n \right \rfloor$.

Evaluate $\sum_{n=1}^{1024}\left \lfloor \log_2n \right \rfloor$. I thought the answer is $1+1*2+2*2^2+3*2^3+4*2^4+5*2^5+6*2^6+7*2^7+8*2^8+9*2^9+2^{10}=9219$, but the answer should be 8204. What ...
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4answers
274 views

Show that $x \ln(ex) - \sqrt{x}\geq 0$ for all $x\geq 1$

How do I continue to prove this? Show that $$ x \ln(ex) - \sqrt{x}\geq 0 $$ for all $$ x \geq1 $$ My try: $$\begin{eqnarray*} \\ \ln(e^x) + \ln(x^x) &\geq& \sqrt{x} \\ \\ \ln(e^x) ...
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votes
5answers
130 views

Proof $\log_b a\cdot\log_c b\cdot\log_a c=1$,

Please help me proof $\log_b a\cdot\log_c b\cdot\log_a c=1$, where $a,b,c$ positive number different for 1.
5
votes
4answers
2k 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 ...
0
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2answers
909 views

Analytically find the domain of a logarithmic function?

I'm taking pre-calc and I'm already falling behind this semester. I'm hoping someone could give me a simple explanation on how to solve these types of problems: $$f(x) = \log_5(4-x^2)$$ I have the ...
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vote
1answer
83 views

Is this log form simple enough?

$$\frac{3}{\ln{2}-12}$$ Is this form simplified enough? There is a number '$12$' below the fraction line, do i need to transform the $\log$ more to make it simpler? I wrote that in a college math ...
4
votes
2answers
214 views

How should “7 $\log_{10}$” be interpreted?

A cookery related article I want to refer to mentions a "7 log10 relative reduction of salmonella". A few related sources suggest this evaluates to 10,000,000, although I would have imagined that 107 ...
0
votes
2answers
2k views

how to simplify log base 2 and log base 4

How do I simplify the following expression: $$\log_2(2x+1) - 5\log_4(x^2) + 4\log_2(x)$$ That's it, please help me ok?
3
votes
4answers
81 views

Vessels received by the brother monarch

An eastern monarch sends 10.000 golden vessels to a brother monarch, whose kingdom is many days march distant. The gift is carried on camels. Each merchant who supplies camels for some part of the ...
2
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2answers
69 views

Prove this identity

Help me prove $\log\tan 1^0\cdot\log\tan 2^0\log\tan 3^0\cdot\cdot\cdot\log\tan 58^0\cdot\log\tan 59^0\cdot\log\tan 60^0=0$
3
votes
3answers
227 views

Proof the logarithmic identity $\log_{b+c} a+\log_{c-b} a=2\log_{b+c} a\cdot\log_{c-b} a$

Please help me proof $\log_{b+c} a+\log_{c-b} a=2\log_{b+c} a\cdot\log_{c-b} a$, for $a,b,c>0$ and $a^2+b^2=c^2$. Thanks.
1
vote
0answers
41 views

Triangular exponentation logarithm and inverse

The generalized formula of triangular exponentation on real numbers field is $x ^ {\triangle y} = \frac {1} {y \cdot B (x, y)} = \frac {\Gamma(x + y)} {\Gamma(x) \cdot \Gamma(y + 1)} $ It's my ...
1
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3answers
69 views

Proof the logaritmic identity

Please help me proving the basic logarithmic identity $\log_3 12=1+\log_5 4\cdot \log_6 5\cdot \log_3 6$
6
votes
3answers
119 views

Solving $\ln(x^2+1)+1 = \ln(x^2+4)$

This is a homework question, but I've tried as hard as I can. Let me walk you through what I've done so far. $$\ln(x^2+1)+1 = \ln(x^2+4)$$ $$\ln(x^2+4) - \ln(x^2+1) = 1$$ ...
3
votes
4answers
428 views

Prove $\log_{\frac{1}{4}} \frac{8}{7}> \log_{\frac{1}{5}} \frac{5}{4}$

Prove $$\log_{\frac{1}{4}} \frac{8}{7}> \log_{\frac{1}{5}} \frac{5}{4}$$ How to prove without a computer?
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votes
3answers
741 views

Evaluating $\int_0^{\large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $

It's my first post here and I was wondering if someone could help me with evaluating the definite integral $$ \int_0^{\Large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $$ Thanks in ...
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1answer
47 views

For what values of $a$ is $\log_a(n)$ is $Big$ $\Theta$ $(\log_2(n))$

I should end up with a range for $a$, but I end up with a single value for $a$ after evaluating $Big$ $O$ and $Big$ $\Omega$. Problem: Prove $\log_a(n)$ is $Big$ $\Theta$ $(\log_2(n))$. For which ...
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2answers
847 views

Express as a single logarithm with a coefficient of 1

Express as a single logarithm with a coefficient of 1: $$ 2(\ln(x)-\ln(x+1))-3(\ln(x^2)-\ln(x^2-1)) $$ I've been trying for nearly an hour and can't seem to find the answer, can anyone help plz :S
4
votes
1answer
122 views

Prove that $\log_{2b+c}a+\log_{2c+a}b+\log_{2a+b}c\ge\frac{3}{2}$

Let $a\ge 3,b\ge3,c\ge3$. Prove that: $$\log_{2b+c}a+\log_{2c+a}b+\log_{2a+b}c\ge\frac{3}{2}$$ I don't know what to do. Rewrite to $\ln(x)$ or $e^x$ But it's not work
0
votes
1answer
85 views

Finding correlation in plotted graph

I have the following sets of values for $x$ and $y$: $x$: $0, 1.5, 2.0, 3.0, 5.0$ $y$: $0.00, 0.92, 1.41, 2.60, 5.59$ I am to find a correlation between the two sets of values. A graph of them ...
0
votes
0answers
52 views

Finding correlation through plotting logarithms

I have a problem in which I am to find the correlation between two sets of data. Plotting them as logarithms makes for a constant increase, except for one of the data points (the first), as seen in ...
2
votes
1answer
672 views

Using the integral definition [duplicate]

Possible Duplicate: Natural Logarithm and Integral Properties I was asked to prove that ln(xy) = ln x + ln y using the integral definition. While I'm ...
2
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2answers
96 views

Are fractional exponents considered logarithms?

Say I have a number with a fractional exponent, $10^{\frac{1}{3}}$. Could this number be considered a logarithm, even though it is not written as $10^{0.\overline{3}}$?
5
votes
3answers
635 views

Verifing $\int_0^{\pi}x\ln(\sin x)\,dx=-\ln(2){\pi}^2/2$

I used all I know to show that $$\int_0^\pi x\ln(\sin x)dx=-\ln(2) \pi^2/2$$ This is my homework but don't know where to start. I appreciate your help.
1
vote
2answers
103 views

Simplify $\ln(|x-x^2|) - \ln(|x-1|)$

As the topic says, I need to simplify: $$\ln |x-x^2| - \ln |x-1| $$ I don't know how to approach the problem at all. I'm not asking for the answer, but something to maybe get me going.
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3answers
71 views

What is the max possible value of $2i-2^{\lceil{\log{i}}\rceil}$ when the base of log is 2?

I have an expression of the form $2i-2^{\lceil{\log{i}}\rceil}$ . I want to know the maximum value of $2i-2^{\lceil{\log{i}}\rceil}$ . Please consider the base of the logarithm as 2.
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votes
1answer
180 views

How to find asymptotic entire functions?

I want to know how to find analytic functions $f(z)$ that are asymptotic and analytic on and near the real line of functions of the type $\ln(C +\exp(P(z^2)))$ where $C$ is a complex constant and $P$ ...
3
votes
2answers
81 views

How does this logarithm transformation works?

I am reading this page about logarithm: http://www.andrews.edu/~calkins/math/webtexts/numb17.htm And saw this piece of transformation: $$\log_b(\frac{2x^2 + 2x}{12}) = 0$$ Take exponents of both ...
10
votes
5answers
713 views

Solve the equation $2^x=1-x$

Solve the equation: $$2^x=1-x$$ I know this is extremely easy and I know the solution using graphical approach. Basically, I can see the solution, but I can't work it out algebraically.
2
votes
1answer
448 views

Definite integral involving e and ln

I am supposed to solve a definite integral which involves $\ln$ and $e$. Nowhere in my textbook can I even find examples of how this would be done. I do know that $\ln(e(x))$ and $e(\ln(x))$ ...
3
votes
1answer
403 views

How to find a summation of a logarithmic function?

Suppose that I had to find $\log_{10}(8952!)$. Now, since $\log(a) + \log(b) = \log(ab)$, this can be rewritten to the following summation: $$\sum_{x=1}^{8952}{\log_{10}(x)}$$ Would there be a ...
2
votes
2answers
710 views

limits of logarithm

I am trying to understand the definition of a logarithm, because when I was trying to find the derivative of $2^x$ I got $$2^x \lim_{h \to 0} \frac{2^h-1}{h}$$ which I have found by searching to be ...
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1answer
177 views

The Way to solve an equation involving logarithms

How can I solve the following equation: $$2^{log_3 x}+x^{log_3 2}=4$$ I don't want the final answer, I want to know how I can solve these kind of equations.
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5answers
191 views

Limit of a recursively defined bivariate function.

Let m and n be positive integers. Let $f(m,0)=m$ Let $f(m,n)= e \ln(f(m,n-1))$ $$\lim_{m\to\infty} \ln(m)\Big(f(m,\lfloor\ln m\rfloor)) - e\Big) = 163^{1/3}+C$$ Where $C$ is a constant. It seems ...
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votes
4answers
2k views

How to prove $O(\log n)$ is true for a binary search algorithm?

I have already looked at the answer here. I'm trying to understand how the poster got: f(n) = O(1) = O(nlogba) So far I have O(1) = T(n) - T(n/2). How is it that this became O(nlogba) ? EDIT: ...
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vote
1answer
71 views

Use identities to simplify $lg(a^2+b^2)^2$

Use logarithmic identities to simply the following: $$lg(a^2+b^2)^2$$ I started with \begin{eqnarray} lg(a^2+b^2)^2&=&2 \cdot lg(a^2+b^2) \\ \end{eqnarray} I think it's not the final ...
3
votes
1answer
400 views

Chebyshev's first $\vartheta(x)$ function question

This was an exercise using the first Chebyshev function, $\vartheta(x)= \sum_{p \leq x} \log p.$ The question is simply how to prove (2) below, the rest is my two thoughts on how to proceed. [Edit: ...
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votes
2answers
110 views

How ${\left(\frac 12 \right)}^{{\lg n}}$ = ${\frac 1n}$ for any natural number $n$?

Consider binary logarithm . How is the value of ${(\frac 12 )}^{{\lg n}}$ = ${\frac 1n}$? I was going through this video of skiplists and the professor at 53:22 seconds make this claim .
2
votes
4answers
196 views

What is the $(\lg n)$-th root of $n$?

I am looking for the answer of the $(\lg n)$-th root of $n$, that is, $\sqrt[\lg n]{n}$. What is the answer and what log property should I use here? Please assume base as $2$ and $n$ as a natural ...
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votes
3answers
990 views

How can I solve for $n$ in the equation $n \log n = C$?

Believe it or not, this isn't homework. It's been many years since grade school, and I'm trying to brush up on these things. But my intuition isn't helping me here.
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vote
1answer
401 views

Finding the value of y in terms of x.

Is it possible to get the value of $y$ in terms of $x$ from the below equation? If so please give give me a clue how to do that :) $$y \sqrt{y^2 + 1} + \ln\left(y + \sqrt{y^2 + 1}\right) = ...
0
votes
3answers
254 views

Maple Error on Asymptotic Analysis of $\ln(n)!$

In Maple, the command asmypt($f$,$x$) computes the asymptotic expansion of the function $f$ with respect to the variable $x$ (as $x \rightarrow \infty$). The command asympt(ln(n)!,n); gives the ...
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8answers
850 views

Why does the logarithm require a special notation?

Since the logarithm is the reversed exponentiation, why does it need a distinct notation for it? Why can't we just ask: $$2^x=8$$ Instead of: $$\log_2 8=x$$
0
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2answers
78 views

solving in x involving both exponential and logarithmic function

Is it possible to solve a function with both exponential and logarithm such as $a x^2−b.\log(x)= c$ in closed form; where $a,b,c$ are constants and $a>0$ and $b>0$?