Questions related to real and complex logarithms.

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6
votes
2answers
229 views

Why isn't $\frac{\mathrm{d} }{\mathrm{d} x} \ln(x)$ specified as $\frac{1}{x},x>0$?

As I understand, $\begin{eqnarray} \frac{\mathrm{d}}{\mathrm{d}x}\ln(x)\end{eqnarray} $ is generally specified as $\begin{eqnarray} \frac{1}{x} \end{eqnarray}$. Would it be more appropriate to state ...
6
votes
3answers
359 views

Definite integral involving logarithm of cosine

Does anyone know the provenance of or the answer to the following integral $$\int_0^\infty\ \frac{\ln|\cos(x)|}{x^2} dx $$ Thanks.
6
votes
3answers
113 views

Finding $\int_0^{\pi/4}\sqrt{1+\left( \tan x\right)^2}dx$

I would like to understand all the steps to find out this integral $$ \int_0^{\pi/4} \sqrt{1+\left( \tan x\right)^2} dx$$ Wolfram Alpha returns: $$ \frac12 \log(3+2 \sqrt2) = 0.881373587019543...$$ ...
6
votes
3answers
120 views

How prove this $H_{2n}-H_{n}+\frac{1}{4n}>\ln{2}$

Show that, for every positive integer $n$, $$\dfrac{1}{n+1}+\dfrac{1}{n+2}+\cdots+\dfrac{1}{2n}+\dfrac{1}{4n}>\ln{2}$$ I know this ...
6
votes
2answers
3k views

Intuition behind logarithm inequality: $1 - \frac1x \leq \log x \leq x-1$

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 ...
6
votes
3answers
161 views

Evaluate $\int_0^1 {\ln(1+x)\over x}\,dx$.

How would one evaluate $\int_0^1 {\ln(1+x)\over x}\,dx$? I'd like to do this without approximations. Not quite sure where to start. What really bothers me is that I came across this while reviewing ...
6
votes
6answers
237 views

Inequality, what is wrong with $\log(-1) = - \log(-1)$?

Can anyone tell me what is wrong with the following line of argument: $$ \log(-1) = \log(-1) - \log(1) = - \bigg( \log(1) - \log(-1) \bigg) = - \log \Big( \frac{1}{-1} \Big) = - \log(-1) $$ ...
6
votes
2answers
210 views

Derivative of ${ x }^{ x }$ without logarithmic differentiation

With logarithmic differentiation, it is quite simple to compute the derivative of $x^x$: $$y=x^x$$ $$\ln {y} =x \ln{x}$$ $$\frac {1}{y} \frac {dy}{dx} = \ln{x} +1$$ $$\frac {dy}{dx} ...
6
votes
3answers
93 views

Derivative Of $\ln(x)$

It is required to find the derivative of the natural logarithm of $x$: $\frac {d}{dx}\ln(x)$ My solution: Let $f(x)=\ln(x) $ then $f'(x)=\frac {d}{dx}\ln(x) $ By definition:$$f'(x)= \lim_{h\to ...
6
votes
3answers
90 views

Solve $6^{x+8} = 4^{x-1}$

I tried doing $log_6\left(6^{x+8}\right) = log_6{4^{x-1}}$ I got stuck, and I don't think that was the right route.
6
votes
3answers
112 views

How to calculate $\lim_{x \to0} \dfrac{f(x)-f(\ln(1+x))}{x^{3}}$

$f$ is a differntiable function on $[-1,1]$ and doubly differentiable on $x=0$ and $f^{'}(0)=0,f^{"}(0)=4$. How to calculate $$\lim_{x \to0} \dfrac{f(x)-f\big(\ln(1+x)\big)}{x^{3}}. $$ I have ...
6
votes
3answers
263 views

Series comparisons and logarithms?

Prove the convergence of $$ \sum_{n = 1}^{\infty} {\sqrt{\, 2n - 1\,}\,\ln\left(4n + 1\right) \over n\left(n+1\right)} $$ I've been struggling for hours on this. By the textbook we have the limit ...
6
votes
3answers
114 views

What is the value of $\ln \left(e^{2 \pi i}\right)$

I know that $$e^{2 \pi i} = 1$$ so by taking the natural logarithm on both sides $$\ln \left(e^{2 \pi i}\right)=\ln (1)=0$$ however, why isn't this $2 \pi i$ as expected? Is it beacuse logarithms ...
6
votes
2answers
91 views

Is this summation solvable? $S_n = \sum_{i = 1}^{n}\log_i{(n)}$

Is it possible to solve a summation with a variable base of log? $$ S_n = \sum_{i = 2}^{n}\log_i{(n)} $$ Should I use the derivative of $\log_i{(n)}$?
6
votes
3answers
111 views

Need help with logarithmic differentitation

I have the expression $$y = \sqrt{x^2(x+1)(x+2)}.$$ I have tried looking at videos but I still cannot arrive at the correct answer and don't know how to get there. By the way, the correct answer is ...
6
votes
1answer
1k views

How to figure of the Laplace transform for $\log x$?

I was looking at a table of common Laplace transforms of functions when I came across the transform for $\log x$. Apparently, the transform is as follows: $$\mathcal{L} \left\{ \log ...
6
votes
4answers
242 views

What did Johann Bernoulli wrong in his proof of $\ln z=\ln (-z)$?

Some people say, Johann Bernoulli has proven $\ln z=\ln (-z)$ in the following way $$\ln ((-z)^2 )=\ln(z^2)\;\;\;\Rightarrow\;\;\;2\ln(-z)=2\ln z\;\;\;\Rightarrow\;\;\;\ln (-z)=\ln z$$ While the ...
6
votes
3answers
125 views

Series involving log $\sum_{n=1}^{\infty} \left( n\log \left(\frac{2n+1}{2n-1}\right)-1\right) = \frac{1-\log 2}{2}$

Does anybody know how to prove this series? $$\sum_{n=1}^{\infty} \left( n\log \left(\frac{2n+1}{2n-1}\right)-1\right) = \frac{1-\log 2}{2}$$ I arrived at this through Mathematica. I tried writing ...
6
votes
3answers
165 views

Something about $\frac{\log x}{x}$

Denote $\log x = \log_ex$. Let's consider the below function $$\frac{\log x}{x}$$. Apparently, It's maximum is $\frac{1}{e}$. and strictly increasing in $(0,e]$, strictly decreasing in $[e,+\infty)$. ...
6
votes
1answer
87 views

Sum of Logarithm Arguments

This is a very simple question I suspect but I just cannot seem to nail it... I have values for $X,Y,Z $, where $X =\log (x)$, $Y = \log (y)$ and $Z = \log (z)$ and I need to calculate $x + y + z$, ...
6
votes
1answer
126 views

logarithms power equation

I got a home work question to solve the following: $$ 27x^2 < x^{\log_3x} $$ can any one please explain how to solve this type of equation? I have no idea what to do or what to search for.
6
votes
1answer
799 views

Comparing Powers of Different Bases

How can I know if one power is bigger than the other when the bases are different? For example, considering $2^{10}$ and $10^{3}$ the former is the greater one, but how to prove this? Logarithms? ...
6
votes
1answer
131 views

How to compute the mean average exponent of the naturals? What is the limit for large numbers?

With a friend I was trying to get an understanding for why the expected gap between primes is logarithmic. With that motivation I tried to express the average exponent of numbers. By average ...
6
votes
1answer
345 views

Are Base Ten Logarithms Relics?

Just interested in your thoughts regarding the contention that the pre-eminence of base ten logarithms is a relic from pre-calculator days. Firstly I understand that finding the (base-10) ...
6
votes
2answers
135 views

Solving $\ln(x^2+1)+1 = \ln(x^2+4)$

This is a homework question, but I've tried as hard as I can. Let me walk you through what I've done so far. $$\ln(x^2+1)+1 = \ln(x^2+4)$$ $$\ln(x^2+4) - \ln(x^2+1) = 1$$ ...
6
votes
1answer
114 views

$\log^2 (x^2) + \log (x-1) = 0$

I'm trying to solve the equation $\log^2 (x^2) + \log (x-1) = 0$ but all I could do is to show that $1 < x < 2$. Wolfram Alpha says that $x = 1.508554...$, this is good, but I really want to ...
6
votes
2answers
301 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 ...
6
votes
3answers
212 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 ...
6
votes
1answer
87 views

What is $ \lim_{x \to 0} \log_0(x) $?

As per the title; what is $ \lim_{x \to 0} \log_0(x) $ ? According to WolframAlpha: $$ \lim_{x \to 0} \log_0(x) = 0 $$ but how is this possible? Surely the limit should be indeterminate since ...
6
votes
1answer
618 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 ...
6
votes
1answer
259 views

Can I build a program that will tell me if a real world data set looks linear, logarithmic, exponential etc?

I have a bunch of real world data sets and from manually plotting some of the data in graphs, I've discovered some data sets look pretty much logarithmic and some look linear, or exponential (and some ...
6
votes
2answers
46 views

Deriving the analytical properties of the logarithm from an algebraic definition.

Definition: The base $a$ logarithm ($a\in]0,1[\cup]1,+\infty[$) is the continuous function defined by: $\log_a(xy)=\log_a(x)+\log_a(y)~~\forall x,y>0$ and $\log_a(a)=1$ If I used this definition ...
6
votes
2answers
113 views

How to Solve : $ A =\frac{1}{6}\left((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3\right) $

$ A =\frac{1}{6}\left((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3\right).$ Solve for $2^A.$ (no calculators or graphs are permitted) The way I went about solving this problem was using ...
6
votes
1answer
196 views

$\exp(\ln(x))=x$ and $\ln(\exp(y))=y$.

Let $(A,1_A,|\cdot{}|)$ be a unital Banach algebra, for instance $A=M_n(\Bbb R)$ or $M_n(\Bbb C)$. What is the union of all open unit balls $B_{\|\cdot{}\|}$ where $\|\cdot{}\|$ ranges over all ...
6
votes
2answers
363 views

Formula for Sum of Logarithms $\ln(n)^m$

As you know $\sum_{n=1}^k \ln(n) =\ln(k!)$ is there a formula for $\sum_{n=1}^k \ln(n)^m$?
6
votes
1answer
273 views

Closed form $\sum_{n=2}^{\infty} \frac{1}{\ln^n{n}}$ and $\sum_{n=2}^{\infty} \frac{n}{\ln^n{n}}$

Apologies if this has been asked before, but I was playing around with Wolfram Alpha and got approximations but not closed forms for $$\sum_{n=2}^{\infty} \frac{1}{\ln^n{n}} \approx 3.2426094109 $$ ...
6
votes
1answer
107 views

Complex exponent properties?

Here is a line in a proof in a complex analysis text: $\sqrt{1-z^2}=\sqrt{1-z}\sqrt{1+z}$ I know you can't do this in general, but when can you do it? Here is what I tried: ...
6
votes
1answer
280 views

Complex Logs and Roots of Unity

I need to find all the solutions to the following using logarithms: $(e^z-1)^3=1$ where z is a complex number. I am told that using roots of unity I can break this equation down but I must be missing ...
6
votes
1answer
147 views

Differentiating $y=x^x$ with the formal definition of a derivative

A friend and I were messing around with derivatives, and while we both know the procedure for finding the derivative of $y=x^x$ with logarithmic differentiation, i.e. $$y=x^x\\ ln(y)=x*ln(x)\\ ...
5
votes
6answers
1k views

Alternate proof for “$\log_{10}{2}$ is irrational”

I need to prove that $\log_{10}{2}$ is irrational. I understand the way this proof was done using contradiction to show that the even LHS does not equal the odd RHS, but I did it a different way and ...
5
votes
7answers
798 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 ...
5
votes
7answers
3k views

Is there any significance to the logarithm of a sum?

Many years ago, while working as a computer programmer, I tracked down a subtle bug in the software that we were using. Management had dispaired of finding the bug, but I pursued it in odd moments ...
5
votes
2answers
108 views

Evaluate $\lim\limits_{y\to 0}\log(1+y)/y$ without LHR or Taylor series

Problem 12 on p. 297 of Spivak's Calculus (first edition) is Find $\lim\limits_{y\to 0}\log (1+y)/y$. (You can use L'Hospital's rule, but that would be silly.) I'm not sure the other method ...
5
votes
5answers
270 views

How to evaluate $\int_0^1 \frac{\ln(x+1)}{x^2+1} dx$

This problem appears at the end of Trig substitution section of Calculus by Larson. I tried using trig substitution but it was a bootless attempt $$\int_0^1 \frac{\ln(x+1)}{x^2+1} dx$$
5
votes
6answers
248 views

Prove that for all $x>0$, $1+2\ln x\leq x^2$

Prove that for all $x>0$, $$1+2\ln x\leq x^2$$ How can one prove that?
5
votes
3answers
542 views

Basic Logarithm Equation

$\log_2(x) = \log_x(2) $ Using the change of base theorem: $\dfrac{\log(x)}{\log(2)} = \dfrac{\log(2)}{\log(x)}$ Multiplied the denominators on both sides: $\log(x)\log(x) = \log(2)\log(2)$ I kind ...
5
votes
4answers
488 views

The caret ^ symbol means exponentiation informally in math. Why not a symbol for log too? [closed]

Plus, minus, multiply, divide, and exponentiation all have symbols in math (+, -, *, /, ^ ) . But why isn't there the missing log symbol too? Here's how it would work: 4 ^ 5 = 1024 (as is standard ...
5
votes
4answers
150 views

Can someone explain why $x^{\log(a)} = a^{\log(x)}$?

I'm trying to see why the below is true. $$ x^{\log(a)} = a^{\log(x)} $$ Anyone here know why this is? Thank you.
5
votes
5answers
454 views

L'Hôpital's as $x$ tends to infinity

I'm searching for the explanation to the limit of: $$ \lim\limits_{x\to\infty} x\, \ln\frac{x+1}{x-1}. $$ I know the answer is 2, but I can't seem to get there. The problem is in my textbook under a ...
5
votes
5answers
767 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 ...