Questions related to real and complex logarithms.

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5
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1answer
381 views

Inverse function of $\operatorname{li}(x)$ over $x>\mu$?

How can I get the inverse function of $\operatorname{li}(x)$ over $x>\mu$? Where $$\operatorname{li}(x)=\int_{0}^{x}\frac{ds}{\ln(s)}$$ is the so-called logarithmic integral, and ...
5
votes
1answer
120 views

Understanding a famous proof by Jitsuro Nagura: Need help understanding one step in the main theorem

I am going through the proof by Jitsuro Nagura which shows that there is always a prime between $x$ and $\frac{6x}{5}$ where $x \ge 25$. Nagura uses the following definitions: $$\vartheta(x) = ...
5
votes
1answer
144 views

Derivative of $f(x)^{g(x)}$ at points when $f(x)=0$

I am interested in understanding the general behavior of the derivative for $$f(x)^{g(x)}$$ at points where $f(x)=0$. For example, if $f^g=x^n$ we have $$\frac{d}{dx}f^g(0)=\begin{cases}0 & n\ge ...
5
votes
0answers
793 views

Choosing the branch of a logarithm

The problem: I am integrating complex logarithms over an angle $\phi$ over $[0,2\pi]$. It is quite complex (pun not intended) and I called Mathematica in to aid me. I am calculating an energy of a ...
5
votes
1answer
247 views

Series involving Logs

I'm trying to find the name of, and a good online reference to, a type of "logarithm series", e.g. $$(1+x)^9 = \sum_{k=0}^{\infty} \frac{9^k\ln^k(1+x)}{k!} $$ I realise that this comes from $x^y ...
4
votes
4answers
307 views

Clarify why all logarithms differ by a constant

One of the rules of logarithms that I've been taught is: $\log_b(x)$ is equivalent to $\frac{\log_k(x)}{\log_k(b)}$. Recently I've also seen another rule that says: $\log_b(x)$ is equivalent to ...
4
votes
7answers
625 views

Prove that $5/2 < e < 3$?

Prove that $$\dfrac{5}{2} < e < 3$$ By the definition of $\log$ and $\exp$, $$1 = \log(e) = \int_1^e \dfrac{1}{t} dt$$ where $e = \exp(1)$. So given that $e$ is unknown, how could I ...
4
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4answers
147 views

Differentiate $\log_{10}x$

My attempt: $\eqalign{ & \log_{10}x = {{\ln x} \over {\ln 10}} \cr & u = \ln x \cr & v = \ln 10 \cr & {{du} \over {dx}} = {1 \over x} \cr & {{dv} \over {dx}} ...
4
votes
3answers
482 views

Solving $xe^{-x}+2e^{-x}=0$

While I was studying my maths book, I came across this equation: $$ xe^{-x}+2e^{-x}=0 $$ I tried to solve it in different ways, but each time I break up some rule. My best try was this: Let's ...
4
votes
5answers
387 views

Am I allowed to apply L'Hospital's Rule inside of the natural logarithm function?

I have the following limit: $$\lim_{x\rightarrow \infty} \ln\left(\frac{2x^2+1}{x^2+1}\right)$$ If I was finding the limit of only the terms inside the natural log function, I would have the ...
4
votes
5answers
70 views

What is the limit of $\log_k(k^a + k^b)$ for $k \to +\infty$?

I'm not very good with analysis (I never studied it) but because of my "work" on other topics of mathematics I came to this problem. $$\lim_{k \to +\infty }\log_k(k^a + k^b)=\max(a,b)$$ I'm sure ...
4
votes
6answers
150 views

How to calculate $\ln(x)$

I know only to calculate $\ln()$ using a calculator, but is there a way to calculate it without calculator: for example: $\ln(4)= ??$ as far as I know the only way to do so is to draw the graph of ...
4
votes
7answers
244 views

Solving $x^{\log(x)}=\frac{x^3}{100}$

How do I find the solution to: $$x^{\log(x)}=\frac{x^3}{100}$$ So I multiplied 100 both sides getting: $$100x^{\log(x)}=x^3$$ Now what should I do?
4
votes
3answers
358 views

Solving exponential equations using logarithms

This is the equation that I am having troubles with: $$\large x^{\large\log_{10}5}+5^{\large\log_{10}x}=50$$ So the first thing I do, I logarithm the whole expression with $\log_{10}$. So I get: ...
4
votes
4answers
265 views

Can anybody explain how these logarithms are transformed?

I'm learning for my algorithms exam and I can't derive two logarithm transformations: $ 3^{log_{4}(n)}=n^{log_{4}(3)} $ $ log_{3}(n)=log_{3}(e)*ln(n) $ I'm not very strong in logarithms, anybody ...
4
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8answers
234 views

Find $\lim_{x \to 0} \left( \frac{\ln (\cos x)}{x\sqrt {1 + x} - x} \right)$ efficiently

I need to evaluate: $$\lim_{x \to 0} \left( \frac{\ln (\cos x)}{x\sqrt {1 + x} - x} \right)$$ Now, it looked to me like a classic L'Hospital's Rule case. Indeed, I used it (twice), but then things ...
4
votes
4answers
132 views

Solve these equations simultaneously

Solve these equations simultaneously: $$\eqalign{ & {8^y} = {4^{2x + 3}} \cr & {\log _2}y = {\log _2}x + 4 \cr} $$ I simplified them first: $\eqalign{ & {2^{3y}} = ...
4
votes
4answers
1k views

Find the domain of $f(x)=\ln(3x+2)$

Find the domain of $f(x)=\ln(3x+2)$ I can find domain, but is it the same for a $\log$ function? And also, do I have to rid the equation of the $\ln$ before I can find the domain? I'm really ...
4
votes
2answers
313 views

How can we solve: $\sqrt{x} - \ln(x) -1 = 0$?

How could we solve $$\sqrt{x} - \ln(x) -1 = 0$$ without using Mathematica? Obviously a solution is $x = 1$, but what are the other exact solutions? This question is inspired by my first question How ...
4
votes
3answers
729 views

How popular and used were logarithm tables?

I've heard that, for a time, logarithm tables "sold more than the Bible". Can someone produce some reliable documentation about how prevalent they were ? Would a common shopkeep have one ? Would a ...
4
votes
2answers
316 views

Double integral application

I need to determine $$\int_{0}^{1} \int_{-\sqrt{x}}^{\sqrt{x}}\frac{1}{1-y}dydx$$ I integrate in terms of the y component and obtained: $$\int_{0}^{1}\ln(\frac{1+\sqrt{x}}{1-\sqrt{x}})dx$$ Can ...
4
votes
2answers
186 views

Limit of logarithms without l'Hospital

This is my first post so I hope you forgive any formatting mistakes. This is a task out of a training exam, I may add that we have not yet introduced l'Hospital or derivatives. We have to determine ...
4
votes
2answers
492 views

Logarithm as limit

Wolfram's website lists this as a limit representation of the natural log: $$\ln{z} = \lim_{\omega \to \infty} \omega(z^{1/\omega} - 1)$$ Is there a quick proof of this? Thanks
4
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2answers
12k views

The difference between log and ln

$$\dfrac{1}{2}\ln(x+7)-(2 \ln x+3 \ln y)$$ Our professor let's us solve this but i do not understand how $\ln$ works. He says it has same properties with $\log$ but i still don't get it. What's the ...
4
votes
4answers
299 views

Does $\log _b \left( x \right) = \log _b \left( y \right) \rightarrow x = y$?

I hit a snag whilst revising some log rules, could anyone confirm my suspicion: $$\log _b \left( x \right) = \log _b \left( y \right) \rightarrow x = y ?$$
4
votes
2answers
138 views

Solve $ \left( \log_3 x \right)^2 + \log_3 (x^2) + 1 = 0$

I'm new to logarithms and I am having trouble solving this equation $$ \left( \log_3 x \right)^2 + \log_3 (x^2) + 1 = 0.$$ How would I solve this? A step-by-step response would be appreciated. ...
4
votes
3answers
106 views

Why is $x^{\log_x n}=n$?

I'm currently doing a couple of exercises on logarithmic expressions, and I was a bit confused when presented with the following: $5^{\log_5 17}$. The answer is $17$, which is the argument of the ...
4
votes
2answers
191 views

Homework with logarithms

I'm stuck on continuing the next exercise: Considering: $\log_{c}a = 3$ $\log_{c}b = 4$ and: $$ y = \frac{a^{3}\sqrt{b \cdot c^{2}}}{2} $$ What's the value of $\log_{c}y$ ...
4
votes
6answers
107 views

$\lim_{n\to\infty}\left(1+\frac{3}{n}\right)^\frac{n}{2}$

I am trying to resolve this to number $e$. However, I would like to do it in the simplest form. just a note I already tried wolfram but I would like someone to give me a simpler solution. ...
4
votes
2answers
2k views

Intuition behind logarithm inequality

One of fundamental inequalities on logarithm is: $$ 1 - \frac1x \leq \log x \leq x-1 \quad\text{for all $x > 0$},$$ which you may prefer write in the form of $$ \frac{x}{1+x} \leq \log{(1+x)} \leq ...
4
votes
4answers
6k views

Units of a log of a physical quantity

So I have never actually found a good answer or even a good resource which discusses this so I appeal to experts here at stack exchange because this problem came up again today. What happens to the ...
4
votes
2answers
135 views

Is $2^{\log_2(-5)}$ defined?

As far as I know for $\log_2 x$ to be defined $x$ must be higher than 0. However when I enter $2^{\log_2(-5)}$ into wolframalpha it gives result $-5$. Is it mistake?
4
votes
1answer
88 views

A logarithmic equation?

$$\mathcal{O} \left(3^{\log_2(n)} \right) = \mathcal{O} \left(n^{\log_2(3)} \right)$$ Does anyone have any idea how the right side was arrived at? (The $\mathcal{O}$ is Big-$\mathcal{O}$ notation)
4
votes
1answer
198 views

Name for logarithm variation that works on non-positive values?

I've come up with the following variation of a logarithm, intended to work on values that can be 0, or can grow exponentially from zero in either positive or negative direction. $$myLog(x) = ...
4
votes
2answers
54 views

Solve numerical system of nonlinear equations?

I need to solve a nonlinear system of equations that looks like this ...
4
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3answers
162 views

$\log(n)$ is what power of $n$?

Sorry about asking such an elementary question, but I have been wondering about this exact definition for a while. What power of $n$ is $\log(n)$. I know that it is $n^\epsilon$ for a very small ...
4
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4answers
162 views

Prove $\ln{(\frac {x}{y})} = \ln{x} - \ln{y}$ using the definition $\int_1^x\frac{1}{t}dt = \ln{x}$.

Prove $\ln (\frac{x}{y}) = \ln{x} - \ln{y}$ using the definition $\int_1^x\frac{1}{t}dt = \ln{x}$. I am able to prove $\ln{xy} = \ln{x} + \ln{y}$, and $\ln{x^r} = r\ln{x}$, but with this one, I am ...
4
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4answers
146 views

How to find the limit of $\dfrac{\ln(\ln(\frac{n}{n-1}))}{\ln(n)}$?

How do you find $$\lim_{n \to\infty} \dfrac{\ln(\ln(\frac{n}{n-1}))}{\ln(n)}$$ I know it's $-1$, but I had to plot it.
4
votes
1answer
444 views

Exponential and log functions compose to identity

How to prove that the exponential function and the logarithm function are the inverses of each other? I want it the following way. We must use the definition as power series, and must verify that all ...
4
votes
1answer
47 views

$e$ and natural logarithms

How would you solve $6xe^{2x}+3e^{2x}=0$ for $x$ I tried: $\ln(e^{2x})=\ln(1/6x+3)$ $2x=\ln(1)-\ln(6x+3)$ $2x=-\ln(6x+3)$ but then I am stuck there. What am I missing?
4
votes
3answers
101 views

Does $\operatorname{Log}(1+i)^2 =2\operatorname{Log}(1+i)$

And similarly, does $\operatorname{Log}(1-i)^2=2\operatorname{Log}(1-i)$? If we were dealing with real numbers, it would hold. But I'm guessing that the fact that there are imaginary numbers involved ...
4
votes
2answers
132 views

Limiting value of $\lim \frac{1}{k}\sum_{n=1}^k \frac{p(n+1)-p(n)}{\log p(n)}$

Empirically it seems $$\lim_{k\to \infty} \frac{1}{k}\sum_{n=1}^k \frac{g(n)}{\log p(n)} = 1\tag{1} $$ in which p(n) is the nth prime and g(n) is the prime gap $p(n+1)-p(n).$ Cramer conjectured ...
4
votes
2answers
295 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) • ...
4
votes
3answers
176 views

How to prove this ln inequality?

I have the following inequality, which (supposedly) holds for every $x\in\mathbb{R}$: $$ 1+x\ln\left(x+\sqrt{1+x^{2}}\right)\geq\sqrt{1+x^{2}} $$ I've been struggling to find some known inequalities ...
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2answers
745 views

What does $\log^{2}{x}$ mean?

What is it used for and why doesn't it equal $\large\log{x^2}$?
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1answer
92 views

Does the logistic function really uniquely satisfy this?

It is said that the logistic function (denoted $y(u)$ below) is derived from the relation: $$\frac{dy}{du}=y(u)(1-y(u))$$ Does $y(u)=\frac{1}{1+e^{-u}}$ really uniquely satisfy this? I don't see ...
4
votes
1answer
84 views

Prove this polylogarithmic integral has the stated closed form value

Question. Prove the following polylogarithmic integral has the stated value: $$I:=\int_{0}^{1}\frac{\operatorname{Li}_2{(1-x)}\log^2{(1-x)}}{x}\mathrm{d}x=-11\zeta{(5)}+6\zeta{(3)}\zeta{(2)}.$$ ...
4
votes
3answers
85 views

Difficult Integral Involving the $\ln$ function

Please help me solve this integral! I have tried multiple different procedures for integration by parts, as well as substitution and have not come up with anything. $$\int\frac{\ln x}{(\ln x+1)^2}dx$$ ...
4
votes
4answers
114 views

Simple looking log problem

How would I solve this for $x$? The original problem is $$x+x^{\log_{2}3}=x^{\log_{2}5}$$ I have tried to reduce it down to this, $$x^{\log_{10}3}+x^{\log_{10}2}=x^{\log_{10}5}$$ I have been ...
4
votes
2answers
71 views

How to Solve for Zero

$$4x^2e^{-x^2}-2e^{-x^2}=0$$ I took out a common factor of $2e^{-x^2}$ which got me to: $2e^{-x^2}(2x^2-1)=0$ I'm not sure if taking out the common factor helped at all and I don't know where to go ...