Questions related to real and complex logarithms.

learn more… | top users | synonyms

6
votes
4answers
205 views

Proof of $\log_xy=\frac{\log_zy}{\log_zx}$

Why is $\log_xy=\frac{\log_zy}{\log_zx}$? Can we prove this using the laws of exponents?
6
votes
3answers
207 views

What operations on an equation cause it to be destroyed?

I approached my calculus professor about something he said which didn't make much sense to me - He says that in the process of calculating $\lim_{x\to\infty} f(x)^{g(x)}$, you can convert it to ...
6
votes
9answers
119 views

Find the value of $\log_8 9 \times \log_9 10 \times \cdots \times \log_n(n+1) \times \log_{n+1}8$

I'm completely lost on this question. I've been Googling around to no success. Find the value of $$\log_8 9 \cdot \log_9 10 \dotsm \log_n(n+1) \cdot \log_{n+1}8$$ I'm completely stumped as to ...
6
votes
3answers
3k views

Is putting absolute values around the argument of a log obtained through integration incorrect?

I've always been taught that when integrating a function of the form $f'(x)/f(x)$ to put an absolute value around the argument of the resulting logarithm. For example: $$\int\frac1{x}\mathrm dx = ...
6
votes
5answers
81 views

Solve for $x$ : $\log_3(3x + 2) = \log_9(4x + 5)$

Solve for $x$ $$ \log_3(3x + 2) = \log_9(4x + 5) $$ I changed the bases of the logs $$ \frac {\log_{10}(3x + 2)} {\log_{10}(3)} = \frac {\log_{10}(4x + 5)} {\log_{10}(9)} $$ Now I'm stuck, ...
6
votes
3answers
552 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 ...
6
votes
3answers
106 views

Closed-form of $\int_0^1 x^n \operatorname{li}(x^m)\,dx$

I've conjectured, that for $n\geq0$ and $m\geq1$ integers $$ \int_0^1 x^n \operatorname{li}(x^m)\,dx \stackrel{?}{=} -\frac{1}{n+1}\ln\left(\frac{m+n+1}{m}\right), $$ where $\operatorname{li}$ is the ...
6
votes
3answers
1k 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 ...
6
votes
2answers
191 views

Why does my professor say that writing $\int \frac 1x \mathrm{d}x = \ln|x| + C$ is wrong?

My professor says that writing this is convenient $$\int \frac 1x \mathrm{d}x = \ln|x| + C\tag{1}$$ but wrong, since it should be written as: $$\int \frac 1x \mathrm{d}x = \begin{cases}\ln x + C ...
6
votes
3answers
933 views

How to solve $n < 2^{n/8}$ for $n$?

This is from an exercise (1.2.2) in introduction to algorithms that I'm working on privately. To find at what point a $n \lg n$ function will run faster than a $n^2$ function I need to figure out for ...
6
votes
4answers
145 views

Are logarithms the only continuous function on $(0, \infty)$ that has this property?

Are logarithms the only continuous function on $(0, \infty)$ that has this property? $$ f(xy) = f(x) + f(y) $$ If so, how would we show that? If not, what else would we need to show that a function ...
6
votes
4answers
113 views

Solving equation $\log_y(\log_y(x))= \log_n(x)$ for $n$

I'm just wondering, if I log a constant twice with the same base $y$, $$\log_y(\log_y(x))= \log_n(x)$$ Can it be equivalent to logging the same constant with base $n$? If yes, what is variable $n$ ...
6
votes
3answers
190 views

Is $ \log_a x^n = n \log_a x? $

We have: $$\log_a{x^n}=\log_a{\left( x\cdot x \cdot x \cdot ...\cdot x\right)}=\log_a(x)+\log_a(x)+...+\log_a(x)=n\log_a(x)$$ But, $$ \ln\left(-1\right)^2=\ln{\left(-1\right)^2}\\\ln(1)=2\ln(-1)\\ ...
6
votes
2answers
230 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
364 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
4answers
2k views

Summation of logs

Are there any useful identities for quickly calculating the sum of consecutive logs? For example $\sum_{k=1}^{N} log(k)$ or something to this effect. I should add that I am writing code to do this (as ...
6
votes
3answers
122 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
3answers
99 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
6answers
238 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
91 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
277 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
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
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
4answers
245 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
167 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
88 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
128 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
134 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
352 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
116 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
311 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
215 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
631 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
114 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
372 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
274 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: ...