Questions related to real and complex logarithms.

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1answer
70 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 ...
6
votes
2answers
106 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
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4answers
43 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
49 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
49 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
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0answers
65 views

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

I need help evaluating this with contour integration$$ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$$ I am not sure as to how to work with the branch cuts of both $\ln{x}$ and $\sqrt{1-x^2}$ ...
4
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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 $\log_2(z\sqrt{xy})$ ?
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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 $$e^x(x-1)>-2.$$ ...
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2answers
43 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
84 views
+50

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 ...
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votes
1answer
23 views

Indices and law of indices [on hold]

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
56 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
39 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
124 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
57 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
3answers
98 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
275 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: ...
1
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2answers
42 views

How to solve these problems without using L'Hopital's Rule? [closed]

$\lim _{x\rightarrow 0^{+}}\dfrac {\ln \left( \sin x\right) }{\ln \left( \tan x\right) }$ $\lim _{x\rightarrow 0}\left( \dfrac {\sin x}{x}\right) ^{\dfrac {1}{x}}$ $\lim _{x\rightarrow \dfrac {\pi ...
3
votes
2answers
89 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
89 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 ...
3
votes
2answers
88 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
34 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
80 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
48 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
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1answer
31 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
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1answer
26 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
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2answers
71 views

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

Is it correct? $$n^{(\log\,x)} = x^ {(\log\,n)} $$ Can you proof and describe that, for any base? Please explain completely. Thank you.
20
votes
4answers
458 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
70 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
21 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 ...
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1answer
34 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
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2answers
102 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
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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?
1
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2answers
58 views

How does exponentiation relate to multiplication?

My book derives the logarithm function as a definite integral of $1/x$ and defines the exponential function as its inverse. It then extends this definition to other bases: $$b^x = e^{\ln (b) x}$$ ...
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3answers
68 views

Using Riemann sums to show that $\sum_{i=1}^n 1/i = \log{n} +c+O(1/n)$

I want to show that there exists a constant $c$ such that: $$ \sum_{i=1}^n 1/i = \log{n} +c+O(1/n) $$ I am thinking about Riemann sums. Any hints on that?
5
votes
5answers
55 views

Pre-calculus algebra logarithm question

I don't understand how to solve this equation. Been struggling with it and don't know how to start: $$\log_2x=8+9\log_x2$$ Can someone please help me out?
1
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5answers
62 views

Rearrange the equation and solve for $y(x)$ [closed]

How do I solve for $y$? $$ \ln\left(\frac{y-1}{y+1}\right)= x^2 $$
2
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3answers
64 views

Solve $5^{2x+2}-5^{x+2}+6=0 $

How do we solve $5^{2x+2}-5^{x+2}+6=0 $? I know I have to use logarithms but I am not sure how to do it.
2
votes
2answers
55 views

Is it possible to use complex logarithm to integrate $1/(z+i)$ along a path?

Evaluate the following on the path $\gamma_1$ with endpoints $[-1,1+i]$ $$ \begin{align} I_1=\frac{i}{2}\int_{\gamma_1} \frac{1}{z+i}dz -\frac{i}{2}\int_{\gamma_1}\frac{1}{z-i}dz \end{align} $$ Am I ...
0
votes
2answers
55 views

What is the method to correctly isolate $y$ as the dependent variable for $x = e^y$?

In this youtube video about 5:00 minutes in, the instructor makes the point that you can simply exchange the $x$ and $y$ values of the exponential form $x = e^y$ of the equation $y = ln x$ to make $y$ ...
7
votes
3answers
175 views

Logarithm Equality

$$\sqrt{\log_x\left(\sqrt{3x}\right)} \cdot \log_3 x = -1$$ I am not entirely sure how to go about solving for $x$. I cannot square each side because the product isn't $≥ 0$, I can't think of any ...
2
votes
2answers
37 views

Expansion of Logarithms with Cube Roots

Does the following expand to the following $$ \log_6(11^6\sqrt[3]{12}) $$ = $ 6\log_6(11) + \log_6 (\sqrt[3]{12})$
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5answers
125 views

Identity with logarithms?

Is it correct? $$(\log\,n)^{(\log\,n)} = n^ {(\log\,\log\,n)} $$ If yes and they are equal, how can I get $(\log n)^{\log n}$ from $n^{\log \log n}$ ? Thanks.
0
votes
2answers
55 views

Show that $\log[(1+i)^2]\neq 2\log(1+i)$

The problem is as stated in the title. I found that the $\mathrm{Log}[(1+i)^2] = 2\mathrm{Log}(1+i)$. We know that $$\mathrm{Log}(z)=\ln(r)+i\theta$$ Now, without defining a branch, doesn't that mean ...
0
votes
2answers
37 views

Solving equations having both log and exponential forms

How can one Solve equations having both log and exponential forms: For eg... $e^x$ $=$ $\log_{0.001}(x)$ gives $x=0.000993$ (according to wolfram-alpha ...
2
votes
1answer
56 views

Proof of the analyticity of complex logarithm

Let $a\in(-\pi,\pi]$ and $f:G\to\mathbb C$, $G = \{ z\in\mathbb C\setminus\{0\},\operatorname{Arg}z\neq a \}$ $$f(z)=\ln|z|+\imath \arg_a z,\quad a<\arg_az<a+2\pi$$ Prove that $f$ is ...
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votes
2answers
58 views

Is $f(x) = 2 + \ln x$ another way to write $f(x) =\log_e x +2$?

I just want to make sure I am correctly understaning this concept. $f(x) = 2 + \ln x$ is the same as $f(x) =\log_e x +2$ Thus my T graph would look like so: e^y|x+2 -3|2.049 -2|2.135 ...
0
votes
3answers
24 views

How do I rewrite a logarithm in exponential form, so as to plot it? $f(x) = 2\log x$

How do I write $f(x)=2\log x$ in exponential form? Is $2(10)^y=x$ correct?
2
votes
2answers
52 views

Does loga/logb = log(a^(1/logb))?

I know $\log(a^b)=b\log(a)$. However, Wolfram Alpha tells me that $\frac{\log(a)}{\log(b)}$ does not equal $\log(a^\frac{1}{\log(b)})$. Is Wolfram Alpha correct? If it is, why is it correct? I'm ...