Questions related to real and complex logarithms.

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7
votes
2answers
2k views

Value of Summation of $\log(n)$

Context: I am learning Dijstra's Algorithm to find shortest path to any node, given the start node. Here, we can use Fibonnacci Heap as Priority Queue. Following is few lines of algorithm: ...
0
votes
1answer
107 views

How do I solve this exponential equation?

$$x = 2^{x-3}$$ Does there exist an analytical solution to this equation? If so, how do I find it? What if it is changed to an equality? $$x>2^{x-3}$$
16
votes
6answers
474 views

$\log_9 71$ or $\log_8 61$

I am trying to know which one is bigger :$$\log_9 71$$ or $$\log_8 61$$ how can i know without using a calculator ?
3
votes
1answer
157 views

How to prove $\left\|\ln\left(e^{iH_1}e^{iH_2}\right)\right\|\leq\left\|H_1\right\|+\left\|H_2\right\|$?

Let $H_1$ and $H_2$ denote arbitrary Hermitian operators (finite dimensional) and let $\left\|\ldots\right\|$ denote the usual operator norm. I conjecture that $$ ...
0
votes
2answers
146 views

Find all real solutions to the equation $3^{1 + 2\log_3(y-x)} = 48$

$$3^{1 + 2\log_3(y-x)} = 48$$ With this problem I have difficulty getting rid of the exponent. $2\log_5(2y - x - 12) = \log_5(y-x) + \log_5(y + x)$
1
vote
3answers
108 views

Simplify: $3 \ln 2 - \frac{1}{2}\ln 4$

How would you go about simplifying this equation: $$3 \ln 2 - \frac{1}{2}\ln 4$$ I am not very familiar with logarithms and how they work, the process is still confusing me.
0
votes
1answer
392 views

Use the unique factorization for integers theorem and the definition of logarithm to prove that $\log_3 (7)$ is irrational.

Use the unique factorization for integers theorem and the definition of logarithms to prove that $\log_3 (7)$ is irrational. I am taking a beginners fundamental mathematics module, no advanced ...
-2
votes
2answers
191 views

Proving the limit of $\frac{\log(n)^{\log(n)}}{1.01^{n}}$

Can anyone please show me a simple way (if there is one) to show that $$\lim_{n\to \infty}\frac{\log(n)^{\log(n)}}{1.01^{n}}=0$$ And that $$\lim_{n\to \infty}\frac{1.01^{n}}{n!}=0$$ I've checked that ...
11
votes
1answer
363 views

Is the difference of the natural logarithms of two integers always irrational or 0?

If I have two integers $a,b > 1$. Is $\ln(a) - \ln(b)$ always either irrational or $0$. I know both $\ln(a)$ and $\ln(b)$ are irrational.
3
votes
4answers
300 views

Does $\log(x)$ stop at a finite value when x is infinite?

Does $\log(x)$ stop at a certain value when x is infinite? Or is it also infinite? I can see the graph go straighter and straighter in the horizontal direction, and I wonder if it will eventually be ...
1
vote
2answers
145 views

What this graph $x^{\log y}=y^{\log x}$

What is the graph of $x^{\log y}=y^{\log x}$? This question appears on GRE exam.
1
vote
1answer
52 views

Working with logarithms

I have an expression involving very small probabilities $A,B,C,D$. $$x \ge \frac{k_1B}{A + k_2B}$$ $$x \le \frac{k_1C}{D + k_2C}$$ Is there any way for me to check if $x$ satisfies the above ...
10
votes
7answers
705 views

Proving limit with $\log(n!)$

I am trying to calculate the following limits, but I don't know how: $$\lim_{n\to\infty}\frac{3\cdot \sqrt{n}}{\log(n!)}$$ And the second one is $$\lim_{n\to\infty}\frac{\log(n!)}{\log(n)^{\log(n)}}$$ ...
0
votes
1answer
482 views

How to convert log10 values to decimal

I need to convert $\log_{10}(1.07366)$ to decimal. Need the equation for the same.
0
votes
2answers
54 views

Proving an identity involving $f(n) = \sum_1^n{\lceil{\log_2 k}\rceil}$

Prove that $ f(n) = n - 1 + f(\lfloor {n/2} \rfloor) + f(\lceil n/2 \rceil)$ where $f(n) = \sum_1^n{\lceil{\log_2 k}\rceil}$ My trials: At first I find a formula for $f(n)$. If $n = 2^m$ , then ...
2
votes
2answers
52 views

solve $n^{{1/2}^k} = 1$ for $k$

I am trying to find the time complexity for the recurrence $T(n) = 2T(n^{1/2}) + \log n$. I am pretty close to the solution, however, I have run into a roadblock. I need to solve $n^{{1/2}^k} = 1$ for ...
0
votes
1answer
396 views

Try to find an approximation by logarithm function.

Recently I am thinking about this question: Assume $x$ is real, $x\geq0$, $c$ is a positive constant number and $z$ is also a real constant between $3.5$ and $4$. Now there is a function: $$ ...
0
votes
1answer
223 views

Integrating square of derivative of log function

It is well known that $$ \int \frac{f'(x)dx}{f(x)}= \int d \log f(x)=\log f(x) + C $$ In my work I came across the following case: $$ \int \frac{(f'(x))^2dx}{f(x)} $$ I wonder if any interesting ...
1
vote
1answer
30 views

Economics simplification of stochastic transition of capital

I'm taking an macro-econ paper and I can't seem to work out the following simplification. Basically somehow by combining equation 4.14 and ...
1
vote
2answers
101 views

How to reduce the limit one gets when deriving the derivative of the general exponential function?

When applying the definition of a derivative to $\frac{d}{dx}b^x$ and a little algebra one arrives to $$b^x\times\lim\limits_{h \to 0}\frac{b^h - 1}{h}$$ where of course that limit equals $\ln(b)$. ...
1
vote
1answer
347 views

How do you take the partial derivative of a function that has two variables?

I have a speedup function that is a function of two variables: N and P where N is the # of processors and P is the percentage of a computer program that can be parallelized. The speedup formula ...
3
votes
2answers
104 views

How do I use specific data points on a graph to determine an equation?

I need to find a function $f(x)$ such that the following data points would fit on it: $$f(1) = 0 \\ f(2) = 0.5 \\ f(4) = 1.0 \\ f(8) = 1.5 \\ \cdots $$ and so on. So the pattern is every time $x$ ...
6
votes
3answers
321 views

Apparently cannot be solved using logarithms

This equation clearly cannot be solved using logarithms. $$3 + x = 2 (1.01^x)$$ Now it can be solved using a graphing calculator or a computer and the answer is $x = -1.0202$ and $x=568.2993$. But ...
0
votes
1answer
391 views

Linearizing a function using logs

I’m trying to understand how to linearize functions using logs and I can’t quite wrap my head around this. Let’s say we have two functions: p(x) = x^2 q(x) = x(i-1)*1.5 (for x >= 1, q(1) = 1) p: ...
1
vote
2answers
3k views

What is dy/dx of a square root using logarithmic differentiation?

I thought I knew how to do logarithmic differentiation until I came across this: $y = \sqrt{x(x+7)}$ find dy/dx by logarithmic differentiation I was not phased by the problem and did the following ...
4
votes
2answers
379 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 ...
0
votes
4answers
143 views

How does this simplify to $\lg{\sqrt{x}}$?

How does $\lg{x}-\lg{\sqrt[3]{x}}-\lg{\sqrt[6]{x}}$ simplify to $\lg{\sqrt{x}}$? I've tried to get Bagatrix Algebra Solved! to solve it, but it even got the wrong answer... (I checked it by replacing ...
1
vote
1answer
301 views

Problems simplifying logarithmic expressions

I learned all the scripts provided, yet I simply cannot find a way to simplify the following equations: $$4^{log_{2}9}$$ What I've thought so far: 9 may be written as $3^2$, so maybe we could do ...
0
votes
2answers
275 views

Logarithmic Number System Addition

Can somebody explain in simple terms how addition works in a logarithmic number system. Say I have the numbers A and B. These are logarithms (base e, say) of the actual quantities they represent. They ...
0
votes
2answers
312 views

Natural logarithm simplification

$$\begin{align} u(x_1x_2) & = -a\ln{a\over (a+b)x_1} -b\ln {b\over (a+b)x_2} \\ & = a\ln x_1 + b\ln x_2 + \text{constant}. \end{align}$$ Could somebody please explain how this is ...
4
votes
3answers
222 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 ...
1
vote
3answers
916 views

how to get value of n if the value of n * log n is given? [duplicate]

Possible Duplicate: How can I solve for $n$ in the equation $n \log n = C$? how to get value of n if the value of n * log n is given ? I am stuck with this: ...
1
vote
2answers
217 views

Interchanging limits and logarithms

This is probably not too smart, just wondering of the name of this rule: $$ \log \lim_{x \to x_0}f(x) = \lim_{x \to x_0}\log f(x) $$ A reference to a source and/or proof would be good too.
4
votes
3answers
183 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?
0
votes
1answer
131 views

Number of digits in base 10=9+1

Let $\tau:\mathbb N\to\mathbb N$ be the function that counts the number of digits of an nonnegative integer, i.e. $\tau(x)$ is the number of digits of $x$ in base 10. For example $\tau(5)=1$, ...
0
votes
4answers
166 views

How do I find the solution for these exponential equations?

I'm trying to solve these two exponential equations to four decimal places... $4^{5x − 4} = 8$ $(1/8)^x = 85$ But I keep coming up with the wrong answers...help?
1
vote
1answer
213 views

little o notation with natural logs

I'm having trouble with little o notation. Help me show that: $2(n^2 + 100n)\log^5n = o(n^2\sqrt{n})$. It is the last hwk on my sheet and I don't understand it, if someone can help me with ...
3
votes
3answers
87 views

$\log_2$ inequality proof

I would like to know how to prove that : $$ \dfrac{x}{2}< 2^{\lfloor\log (x) \rfloor} \leq x$$ for $x \in \mathbb{N}_+ $ Should I proceed by induction? By contradiction? Thank you
1
vote
1answer
151 views
1
vote
4answers
115 views

Logarithmic expression with three terms

So, I have this logarithmic expression $$\log_5 8-\log_5 20-\log_5 10$$ that I know how to evaluate - the quotient of numbers is the difference of logarithms, so you divide, etc. - but how the heck ...
4
votes
3answers
2k views

Value of Logarithm of negative number

Why the logarithmic value of negative number can't be define? Is there any special reason?
1
vote
1answer
403 views

Composition of logarithm functions

I am a little confused about composition of logarithms, I am given $f(x)=5^{2+5x}$ and $g(x)=\log_5 x$ and I am supposed to find $f(g(x))$. Here is what I have done so far, $$5^{2+5(\log_5 x)}$$ ...
2
votes
2answers
168 views

Finding the inverse of a log problem

I have a homework problem that I am struggling to understand. The problem is Find a formula for the inverse function $f^ {-1}$ of the function $f$. $$f(x)=\log_{2x}3$$ Here is my attempt at solving ...
0
votes
2answers
62 views

products of logarithms

(When one says several things in a question, then several things may get answered and others neglected. Hence this posting overlaps with one of my earlier ones, but (I hope) this one will be short, ...
1
vote
1answer
65 views

Find $\lim_{x\to \infty} \ln(\exp(\operatorname{LmW}(x))+1)(\exp(\operatorname{LmW}(x))+1) - x - \ln(x)$

Find $\lim_{x\to \infty} \ln(e^{\operatorname{LambertW}(x)}+1)(e^{\operatorname{LambertW}(x)}+1) - x - \ln(x)$ Where the $LambertW$ function is defined here : http://en.wikipedia.org/wiki/Lambert_W ...
1
vote
1answer
387 views

Sum of series with log in each term

I was solving recurrence relation of Introduction to Algorithms by {Cormen, Leiserson, Rivest, Stein}, 3rd. edition. Problem 4-3 (i) $$ T(n) = T(n-2) + \frac{1}{lg \; n} $$ I tried few ways, like ...
4
votes
1answer
148 views

What about $\exp((\log x)(\log y))$?

Is anything interesting known about the binary operation $$ x\circ y = \exp_b((\log_b x)(\log_b y)) $$ where $0<b\ne 1$? It's clearly commutatitive and associative, and satisfies $\forall x\in ...
1
vote
1answer
123 views

Big O with Log base equivalences and a question about sum of series.

Hi I need help figuring these out: True or False: $\log_2 n$ is $O(\log_3 n)$ I used the definition of Big O in Dasgupta's book: ${f(n)}\over g{(n)}$ $\leq c$ So I used the base transformation rule ...
2
votes
2answers
61 views

cancelations and logarithms

When faced with the problem of multiplying fractions, for example $$ \frac 5 2 \cdot \frac 8 3\cdot \frac{9}{35} $$ we know that we can permute the numerators, or equivalently, permute the ...
0
votes
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 ...