Tagged Questions

Questions related to real and complex logarithms.

learn more… | top users | synonyms

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
42 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
61 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
46 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
260 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
63 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
250 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
384 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
157 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
120 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
91 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 ?
0
votes
2answers
60 views

what's the relationship with log(sum) and sum(log)?

hi I'm a little confused about the log(sum) function and sum(log) function. In special, what's the relationship between these two terms? $$ -\log \sum_{i}a_i\sum_i b_i $$ $$ -\sum_i\log(a_i+b_i) $$ ...
3
votes
1answer
44 views

Does this graph depict a log scale?

I'm a freelance editor and this graph is in a report and labeled as a log scale. (The version you see is my revision that removes the words "log scale".) The client insists that it is a log scale, ...
3
votes
5answers
147 views

'Proof ' that $\ln(x)$ converges

Where is the flaw in the following 'proof '? $$\lim_{x \to \infty}\left[\frac{\mathrm{d}}{\mathrm{d}x}\left\{\ln(x)\right\}\right]=\lim_{x \to \infty}\left[\frac{1}{x}\right]=0 \implies\lim_{x \to ...
3
votes
3answers
67 views

How to generate $\log$ function that intersects at $(0,1)$ and $(1,0)$?

I apologize for any incorrect or missing formatting, first time posting in the math stack exchange. It's been a few years since I've done any kind of calculus, so I remember nothing at all, which is ...
5
votes
4answers
146 views

Is $\ln\sqrt{2}$ irrational?

I know that the natural log of any positive algebraic number is transcendental, as a consequence of the Lindemann-Weierstrass theorem, but what about the natural log of the square root of two (which ...
4
votes
1answer
282 views

How do I proceed with this integral?

I have the following integral: $$\int \frac{\tan^{-1}(\ln (x))}{x}dx.$$ Trying to solve it by integration by parts (with $u=\ln (x)$ and $v=\tan^{-1} (\ln (x))$, I have seemingly come to a dead end: ...
2
votes
2answers
92 views

How to express $\log_2 (\sqrt{9} - \sqrt{5})$ in terms of $k=\log_2 (\sqrt{9} + \sqrt{5})$?

If $$k=\log_2 (\sqrt{9} + \sqrt{5})$$ express $\log_2 (\sqrt{9} - \sqrt{5})$ in terms of $k$.
3
votes
2answers
93 views

Behaviour of the function $\ln(1+ x^2)$

Thus function has derivative equal to: $\frac{2x}{1+x^2}$. This indicates that it will flatten out while approaching infinity, ie, should have an asymptote. Yet, the function does not have any real ...
4
votes
2answers
111 views

Prove: $\int_{0}^{1}\frac{\ln{x}\,\mathrm{d}x}{\sqrt[3]{x(1-x^2)^2}}\stackrel{?}{=}-\frac18\left[\Gamma{\left(\frac13\right)}\right]^3$

I'd like to evaluate the following definite integral: $$\int_{0}^{1}\frac{\ln{x}\,\mathrm{d}x}{\sqrt[3]{x(1-x^2)^2}}\stackrel{?}{=}-\frac18\left[\Gamma{\left(\frac13\right)}\right]^3.$$ ...
3
votes
1answer
53 views

Prove logarithmic inequality with greatest integer function.

$\left \lfloor n\log_2 n^2 \right \rfloor + \left \lfloor \log_2(\left \lfloor n\log_2n^2 \right \rfloor) \right \rfloor \leq \left \lfloor (n+1)\log_2 (n+1)^2 \right \rfloor + 1$ How to show this? I ...
3
votes
2answers
84 views

What is the meaning of this Wolfram Alpha result when calculating $3^p = 4^q$?

I would like to know are the some $p \in \mathbb{N}$ and $q \in\mathbb{N}$ for $3^p = 4^q$ except the trivial $p = q = 0$. So, I entered the expression into Wolfram Alpha, which returned the result ...
3
votes
1answer
58 views

Rational values of $\sin(\log(x))$

Apart from the trivial solution $\sin(\log(1))=0$, is $$\sin(\log(x))$$ ever rational if $x$ is rational?
1
vote
1answer
39 views

Asymptotics of logarithm: $\frac{1}{n}\ln(a+o(1)) = \frac{1}{n}\ln(a)+o(\frac{1}{n})$

I am having problems with the use of the little oh notation my professor is adopting in the solutions to some exercises. As an example I do not understand why $$ \frac{1}{n}\ln(a+o(1)) = ...
0
votes
1answer
28 views

Subtracting a constant from log-concave function preserves log-concavity, if the difference is positive

I am trying to work out a question from 'Convex Optimization - Boyd' . Specifically, exercise 3.48: Show that if $f : \mathbb R^n \to \mathbb R$ is log-concave and $a > 0$, then the function $g ...
1
vote
2answers
78 views

Is it correct? $n^{(\log\,x)} = x^ {(\log\,n)} $? [closed]

Is it correct? $$n^{(\log\,x)} = x^ {(\log\,n)} $$ Can you proof and describe that, for any base? Please explain completely. Thank you.
24
votes
4answers
776 views

How to find ${\large\int}_1^\infty\frac{1-x+\ln x}{x \left(1+x^2\right) \ln^2 x} \mathrm dx$

Please help me to find a closed form for this integral: $$I=\int_1^\infty\frac{1-x+\ln x}{x \left(1+x^2\right) \ln^2 x} \mathrm dx$$
2
votes
3answers
77 views

If $\ln x$ is defined via an integral and $e$ defined from $\ln x$, how would you prove that $\ln x$ is the inverse of $e^x$?

This is a somewhat technically specific question about the relationship between $\ln x$ and $e^x$ given one possible definition of $\ln x$. Suppose that you define $\ln x$ as $$\ln x = ...
2
votes
1answer
22 views

Find $z$ as a function of $w$ in terms of the complex logarithm, where $w=f(z):=2e^z+e^{2z}$

I have solved the following problem but would like to double check that I did it properly. The problem says: Find an expression for $z$ as a function of $w$ in terms of the complex logarithm, where ...
-1
votes
1answer
35 views

Explanation of the passage from $\int_{N'}^N dN/N$ to $\ln N-\ln N'$

While going through my text I got stuck in the derivation given in the picture. ($\Omega$ is a constant) I don't know how to get the second step from the first step, also I don't know why ln is ...
1
vote
2answers
103 views

Can one use logarithms to solve the equations $2=3^x + x$ and $2=3^x x$?

Could someone explain how would you solve: $$2=3^x + x$$ and $$2=3^x \cdot x$$ I can only solve halfway through. And why is $$10^{\log (x)}= x$$ Thanks
1
vote
1answer
30 views

It's on Indefinite Integrals

$$\int \sqrt{ 1 + 2 \tan x ( \tan x + \sec x )} dx$$ Please tell me the way of solving such questions. like what could i assume sec x or sec x tan x to be equal to?