Questions related to real and complex logarithms.

learn more… | top users | synonyms

1
vote
2answers
22 views

Prove that the binary representation of a number n will use floor(lg(n)) + 1 bits.

I'm taking Computer Algorithms class and one of my problems is from Skiena's Algorithm Design Manual, 2-41: Prove that the binary representation of $n \ge 1$ has $\lfloor \lg n \rfloor +1$ bits ...
0
votes
1answer
21 views

Logarithm Base Question

Suppose you have a integer n. Log2(n) is supposed to be ~ the number of times you have to divide n by 2 until you reach one. Now let's say you want to know ~ the number of times you have multiply n by ...
2
votes
0answers
14 views

Comparing asymptotic growth of logarithmic functions by reasoning

As an exercise, we're sorting functions according to their asymptotical growth. When comparing these two functions, I'm getting stuck: $n^2/(\log_2 n)^3$ versus $n \log_2 n$. Using limits I am ...
0
votes
0answers
27 views

Find the partial derivative with respect to y of the function $f(x,y)=ye^{xy}$

My solution was $e^{xy} + xy e^{xy}$, but when I checked the solution manual it said the answer is $xy e^{xy} \log e + e^{xy}$. So I solved each function for $y$ by setting them each equal to $0$. ...
2
votes
1answer
62 views

Why does $\sqrt{x-x^{\frac{1}{x}^{\frac{1}{x}}}}=\log_{\sqrt{x-x^{\frac{1}{x}^{\frac{1}{x}}}}}(x)?$ (Error)

I might be being very silly here, but I can't for the life of me see why $$\sqrt{x-x^{\frac{1}{x}^{\frac{1}{x}}}}=\log_{\sqrt{x-x^{\frac{1}{x}^{\frac{1}{x}}}}}(x)$$for $x\in \mathbb{Z}, x>1$? ...
0
votes
1answer
35 views

Graphing natural logarithms

I don't know how to obtain the graph of these functions. Could someone please help me? I know what the graph of ln looks like, but other then that I don't know where to go. Thank you for any help ...
0
votes
2answers
36 views

Inequalities with logarithms and limits

For my analysis homework, I am to show that $\lim_{n \to \infty} \frac{3^n}{n!} = 0$ using the epsilon definition. My approach is to invoke the squeeze theorem and show that the above sequence is less ...
3
votes
1answer
30 views

Growth rate of logarithmic function?

Just curious about the growth rate of the logarithmic function: Does there exist a real number $n$ such that $lim_{x \to \infty} \frac{(ln(x))^{n}}{x}$ diverges (does not converge to $0$)? Thanks in ...
0
votes
1answer
35 views

help with logarithmic integration.

I've been googling some tutorials on integrating logarithms for my calc 2 class and I've found a lot of good stuff. Unfortunately nothing has answered how to handle a problem that I have. I've tried ...
0
votes
0answers
45 views

How is $O(\log(\log(n)))$ also $O( \log n)$?

How is $O(\log(\log(n)))$ also $O( \log n)$? I have seen this result somewhere with this but I still didn't quite understand how this is true. This would also help me compute Big Omega of the ...
1
vote
2answers
54 views

Algebraically, how are $-\ln|\csc x + \cot x| +C $ and $\ln| \csc x - \cot x|+C$ equal?

Algebraically, how are $-\ln|\csc x + \cot x| +C $ and $\ln| \csc x - \cot x|+C$ equal? I know both of these are the answer to $\int \csc x \space dx$, and I am able to work them out with calculus ...
-1
votes
3answers
24 views

Solve for two variables, two equations with exponents [closed]

Solve for both k and x, where $5=k(300)^x$ and $80=k(600)^x$
0
votes
1answer
27 views

Solving $1/n^{\lg (n)}$

I am struggling with logarithms and their computation when it comes to computing time complexity. I have a simple complexity: $\frac{1}{n^{\lg (n)}}$, where the logarithm base is 2. How can I reduce ...
23
votes
1answer
236 views

Evaluating $\int_0^{\Large\frac{\pi}{2}}\left(\frac{1}{\log(\tan x)}+\frac{1}{1-\tan(x)}\right)^3dx$

Using the method shown here, I have found the following closed form. $$ \int_0^{\!\Large \frac{\pi}{2}}\!\!\left(\frac{1}{\log(\tan x)}+\frac{1}{1-\tan x}\right)^2\! \mathrm dx= ...
0
votes
0answers
15 views

Exponential and Logarithmic Differentiation.

Q. If $xe^{xy}=y+sin^2x$, then find $\frac{dy}{dx}$ at x=0. If we differentiate the function directly as follows: $e^{xy}+xe^{xy}\left[y+x\frac{dy}{dx}\right]=\frac{dy}{dx}+sin\left(2x\right)$ At ...
2
votes
2answers
54 views

Showing if $n \ge 2c\log(c)$ then $n\ge c\log(n)$

Is this true that if $n \ge 2c\log(c)$ then $n\ge c\log(n)$, for any constant $c>0$? Here $n$ is a positive integer.
34
votes
2answers
888 views

Closed form for ${\large\int}_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx$

Here is another integral I'm trying to evaluate: $$I=\int_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx.\tag1$$ A numeric approximation is: ...
3
votes
2answers
83 views

Why does $\lim_{x\rightarrow\infty} x-x^{\frac{1}{x}^{\frac{1}{x}}}-\log^2x=0?$

Why does $$\lim_{x\rightarrow\infty} x-x^{\frac{1}{x}^{\frac{1}{x}}}-\log^2x=0?$$ Moreover, why is $$x-x^{\frac{1}{x}^{\frac{1}{x}}}\approx\log^2 x?$$
1
vote
0answers
58 views

Solving ${c_1}^x+\sqrt{\frac{\log(x)x}{2}}+3\log(x)x \le c_2$

Is there any way to solve $${c_1}^x+\sqrt{\frac{\log(x)x}{2}}+3\log(x)x\le c_2,$$ for $x>1$, $0<c_1<1$, and $0<c_2<<1$? Thanks
5
votes
2answers
318 views

A logarithmic integral

How can I evaluate following logarithmic integral: $$\int\limits_0^1 \frac{\ln x\ln ( 1 - zx )}{1 - x} dx$$
0
votes
1answer
79 views

How to find numbers $k$ such that $kx - \ln(ex + 1-x) $ is positive on $(0,1]$?

I want to find a condition on $k$ such that $g(x)= kx - \ln(ex + 1-x) > 0$, $x\in [0,1] $. At zero the function is zero. So, to find a condition on $k$ I use $g'(x) > 0$ i.e. $$ k > ...
0
votes
1answer
17 views

Distributing out log equation

$$\log_{27}x = 1 - \log_{27}(x-0.4)$$ $$\log_{27}(x(x-0.4))=1$$ $$x=5.4,\, x=-5$$ I'm confused on the second line. How come it is not $\log_{27}(x+x-0.4)$?
4
votes
1answer
46 views

Simplify an iterated function

If we iterate the function $f(x) = \ln(x + 1)$, we get: $$f(f(x)) = f^2(x) = \ln(\ln(x + 1) + 1)$$ $$f(f(f(x))) = f^3(x) = \ln(\ln(\ln(x + 1) + 1) + 1)$$ $$f(f(f(f(x)))) = f^4(x) = \ln(\ln(\ln(\ln(x + ...
0
votes
2answers
22 views

Simple Logarithmic Question

I have the following equation: $\log(S_n) = \log(u)[2T-n]\,\,$ I was just wondering how $S_n = u^{2T-n}$ is then obtained? Thank You
2
votes
4answers
91 views

Is there any expansion for $\log(1+x)$ when $x\gt 1$?

Is there any expansion for $\log(1+x)$ when $x\gt 1$ ?
0
votes
0answers
39 views

Would the growth rate for base 2 and 10 logs be the same?

Since $\log_{2}(x) = \frac{\log_{10}(x)}{log_{10}(2)}$ and $\log_{10}(2)$ is just a constant, would their growth rate be the same?
0
votes
1answer
16 views

Bounding a logarithmic relation

If I have the following relation $T(n) \le an\lceil \operatorname{lg} (n) \rceil - an +2bn + n$, is it possible to bound $T(n)$ such that it is in the form $T(n) \le an\operatorname{lg}(n) + bn $ for ...
1
vote
2answers
32 views

What log rule was used to simply this expression?

I'm unclear how the left side is equal to the right side. $$365\log(365) - 365 - 305\log(305) + 305 - 60\log(365) = 305\log\left(\frac{365}{305}\right)-60$$ I know $\log(a) - \log(b) = \log (a/b)$ ...
2
votes
1answer
33 views

Having trouble understanding why the $r$-th mean tends to the geometric mean as $r$ tends to zero

I am having trouble understanding the proof of Theorem 3 in "Inequalities" by Hardy, Littlewood and Pólya. This theorem states that the $r$-th mean approaches the geometric mean as $r$ approaches ...
0
votes
1answer
18 views

Domain of a multiple logarithmic function.

Find the domain of the following function: $f\left(x\right)=log_4\left(log_5\left(log_3\left(18x-x^2-77\right)\right)\right)$ My text provides a solution which goes like: => ...
0
votes
2answers
60 views

Prove symmetry of natural logarithm

Prove that $f(x)=\ln\sqrt{x^2+1}$ is symmetrical in $x=0$. $\ln\sqrt{(x-a)^2+1}=\ln\sqrt{(x+a)^2+1}$ $\sqrt{(x-a)^2+1}=\sqrt{(x+a)^2+1}$ $(x-a)^2+1=(x+a)^2+1$ $x^2-2ax+a^2+1=x^2+2ax+a^2+1$ ...
0
votes
1answer
45 views

On the existence/applications of infinitely-nested functions

Inside a previous question, one particular nested function shown is the known tetration. This "kind" of arbitrary repeated functions has always intrigued me, because inside their properties lie so ...
3
votes
3answers
254 views

Find a real entire function $f(z)$ asymptotic to $\ln(x^2+1)$ for real $x$.

Find a real entire function $f(z)$ asymptotic to $\ln(x^2 +1)$ for real $x$. More specific I want $f(0)=0$ and $\frac{1}{2} \ln(x^2+1) < f(x) < 2 \ln(x^2+1)$. Or prove it does not exist.
2
votes
2answers
89 views

Calculating the integral of a logarithmic expression.

The problem I have been working with is $$\int \frac 1{\sqrt x(1+\sqrt x)}\,dx$$ The first step I did to solve this question was to set $u= 1+ \sqrt x$ the set $du = (1/2) x^{-1/2}$ Then I set ...
3
votes
0answers
62 views

Inverse of $x^2+\log^2\cos x$

I'm looking for the inverse of $$f(x)=x^2+(\log\cos x)^2$$ Where $f$ is defined from $[0,\pi/2)$ It dosen't have to be closed form, a sum, an integral or some special functions would be of interest ...
18
votes
1answer
247 views

Integral ${\large\int}_0^1\ln(1-x)\ln(1+x)\ln^2x\,dx$

This problem was posted at I&S a week ago, and no attempts to solve it have been posted there yet. It looks very alluring, so I decided to repost it here: Prove: ...
4
votes
2answers
354 views

Solving a logarithmic expression without a calculator

How do I find the value of this logarithmic expression without using a calculator? I'm trying to relearn algebra, but this problem has me scratching my head, and my Google tutorial searches are ...
3
votes
1answer
47 views

How to get 2 using a standard scientific calculator without pressing the number buttons 0 to 9 and the buttons $+-\times\div$?

I was challenged by a friend to get a number 2 by using a standard scientific calculator but without pressing the number buttons 0 to 9 and the buttons $+-\times\div$. I could get 1 from $\ln e=1$. ...
1
vote
2answers
90 views

solve for $x$ without using softwares $\log_{\sqrt{x}}2+\log_6x^x=4$

Is there any nice way to solve this equation without wolfram? $\log_{\sqrt{x}}2+\log_6x^x=4$ Thanks.
0
votes
1answer
19 views

How do you solve a recurrence with a functin through induction?

I found the answer in part-A by substitution, as O(n) from; T(n/2^k) = T(1).... n/2^k = 1..... so k = 1og2(n)..... T(log2(n)) = T(n/n)+5.... so O(n) IS THE ANSWER, Correct me if am wrong because am ...
7
votes
2answers
153 views

Integral $ \int_{0}^1 \sqrt{\frac{\ln{x}}{x^2-1}} dx$

Please help evaluating this integral $$ \large\int_{0}^1 \sqrt{\frac{\ln{x}}{x^2-1}} dx$$ Mathematica could not evaluate it in a closed form. Numerically it is about ...
2
votes
4answers
45 views

Simplifying/solving a logarithm $\log_24^{2n}$

Need help with simplifying this logarithm. $$\log_24^{2n}$$ Would I just pull the 2n to the front: $$2n*\log_24$$ So it would simplify to $$4n$$ Is this correct or am I completely wrong?
2
votes
1answer
61 views

Generalized Logarithmic Integral - reference request

This page at I&S forum defines the Generalized Logarithmic Integral as $$L\left[ \begin{matrix} a,b,c \\ d,e,f \end{matrix};z\right] =\int_0^z \frac{\log^a x \log^b(1-x)\log^c(1+x)}{x^d (1-x)^e ...
2
votes
1answer
64 views

How does $\left(\log \sqrt x\right)^2 = \frac 14(\log x)^2\;?$

So as the title says it all: How does $\;\left(\log \sqrt x\right)^2 = \frac 14(\log x)^2 \;?$ To be specific, why the removal of root, and how do we get 4 in denominator?
2
votes
0answers
118 views

Contour Integral $ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$

I need help evaluating this with contour integration $$ \int_{0}^{1}{\ln\left(\,x\,\right)\over \,\sqrt{\vphantom{\large A}\,1 - x^{2}\,}}\,{\rm d}x $$ I am not sure as to how to work with the branch ...
3
votes
2answers
20 views

Condensing Fractional Logarithms

Does the following condense to the following: $\log_2z+(\log_2x)/2+(\log_2y)/2 = \log_2(z\sqrt{x}\sqrt{y})$ or to $\log_2(z\sqrt{xy})$ ?
0
votes
2answers
40 views

How to solve exponential inequality with $x$

I need to solve the following inequality. $$\ln(x) - x > 0.$$ I oddly remember that it can only be done by using the graph... Is it true? I have the same problem with $$e^x(x-1)>-2.$$ ...
1
vote
2answers
44 views

Proof of logarithmic identity $\log_g x=\log_a x\cdot\log_g a$

I have to prove the alleged link between the logarithms in base g and a $$\log_g x=\log_a x\cdot\log_g a$$ I know that this can be written as: $$\frac{\ln x}{\ln g}=\frac{\ln x}{\ln g}$$ But does ...
3
votes
1answer
90 views

Base of logarithm decrease when variable count increase

I run a large online platform where users submit articles and earn points. I am working on an algorithm where the more comments they submit, the higher score they will receive. In its simplest ...
-2
votes
1answer
24 views

Indices and law of indices [closed]

Simplify $2^{x+3} + 2^x + 16(2^{x-1})$ in the form $k\cdot 2^x$ , where $k$ is a constant. How to simplify in the form that had given ?