Questions related to real and complex logarithms.

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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 ...
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5answers
214 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 $$\int_0^1 \frac{\ln(x+1)}{x^2+1} dx$$
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6answers
242 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?
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5answers
420 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 ...
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5answers
464 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 ...
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3answers
377 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: ...
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3answers
240 views

Summing up the series $a_{3k}$ where $\log(1-x+x^2) = \sum a_k x^k$

If $\ln(1-x+x^2) = a_1x+a_2x^2 + \cdots \text{ then } a_3+a_6+a_9+a_{12} + \cdots = $ ? My approach is to write $1-x+x^2 = \frac{1+x^3}{1+x}$ then expanding the respective logarithms,I got a series ...
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4answers
242 views

Mathematical notation/name for the number of times a number can be divided by 2

I am using this simple snippet of code, variants of which I have seen in many places: for(int k = 0 ; n % 2 == 0 ; k++) n = n / 2; This code repeatedly ...
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2answers
451 views

How to solve this equation? Can I treat as a quadratic equation?

$$\ln(x+3)+\ln(x-4)=0$$ How to solve this equation? First removing the 'ln' from the equation and after making a quadratic equation and then solve the quadratic equation?
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4answers
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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 ...
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2answers
211 views

Solving $x^2 - 1 = e^x$

Can someone help me solve the equation $x^2 - 1 = e^x$ ? I tried taking the natural logarithm of both sides but I don't know where to go from there.. I got: $\ln(x^2 -1) = x$ But I don't know how ...
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3answers
684 views

Verifing $\int_0^{\pi}x\ln(\sin x)\,dx=-\ln(2){\pi}^2/2$

I used all I know to show that $$\int_0^\pi x\ln(\sin x)dx=-\ln(2) \pi^2/2$$ This is my homework but don't know where to start. I appreciate your help.
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3answers
786 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 ...
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4answers
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Evaluate $\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$ using complex analysis

How do I compute $$\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$$ What I am doing is take $$f(z)=\frac{(\log z)^2}{1+z^2}$$ and calculating $\text{Res}(f,z=i) = \dfrac{d}{dz} \dfrac{(\log ...
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4answers
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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 ...
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3answers
203 views

A little question about logarithm

Can we say that $\log _4 (n^2)=\log _2(n)$? If that is the case, then $\displaystyle 2^{\log _4 (n^2)}=n$? Thanks.
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4answers
300 views

Can we prove $a^{\log_bn} = n^{\log_ba}$?

Can we prove $$a^{\log_bn} = n^{\log_ba}?$$ I forget how to prove this theorem. I picked up one numbers for test, and they worked.
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2answers
222 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 ...
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3answers
258 views

Solving the equation $3^{5x-2}=8^{8x-9}$

I'm trying to solve the equation $$3^{5x-2}=8^{8x-9}.$$ I'm assuming I need to do some work with logarithms, but I don't know what to do. Thanks in advance!
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3answers
181 views

Find the value of $x$ such that $2^x=10$

Given that $\log 5 = 0.7$ (to one decimal place), find the value of $x$ such that $2^x = 10$ (again to one decimal place) I don't know what to do with the information that $10^{0.7} = 5$. Why is ...
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1answer
161 views

Why does $\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$

I read on Wikipedia that $$\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$$ to a high degree of accuracy. Why is this true? Is this pure coincidence or is there some mathematical ...
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2answers
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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 $\log_{c}y$ ...
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3answers
281 views

Does $\log(ab)^n$ equal $(\log(a)+\log(b))^n$ or $n\log(a)+n\log(b)$?

I think this might be a case of slight ambiguity in notation, but here goes: On a test question, I was required to expand the expression $\log (ab)^n$. Since the logarithm is a function, I reasoned ...
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2answers
322 views

A logarithmic integral

How can I evaluate following logarithmic integral: $$\int\limits_0^1 \frac{\ln x\ln ( 1 - zx )}{1 - x} dx$$
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Equation with Logarithm in Exponent

How to I solve the following exercise with a logarithm? I've forgotten the "trick" for doing so: $x^{log_{10} x} =10^4$
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4answers
199 views

To find the logarithm of $1728$ to the base $2 \sqrt{3}$

Find the logarithm of: $1728$ to base $2\sqrt{3}$. Let, $\log_{2\sqrt{3}} 1728 = y$, then $$\begin{align} (2\sqrt{3})^y &= 1728\\ 2^y(\sqrt3)^y &= 1728\\2^y(3^\frac12)^y &= ...
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2answers
174 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} ...
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3answers
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Summation of a series.

I encountered this problem in Physics before i knew about a thing called Taylor Polynomials My problem was that i had to sum this series : ...
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4answers
348 views

What are logarithms?

I have heard of logarithms, and done very little research at all. From that little bit of research I found out its in algebra 2. Sadly to say, I'm going into 9th grade, but yet I'm learning ...
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1answer
136 views

Prove this polylogarithmic integral has the stated closed form value

Question. Prove the following polylogarithmic integral has the stated value: $$I:=\int_{0}^{1}\frac{\operatorname{Li}_2{(1-x)}\log^2{(1-x)}}{x}\mathrm{d}x=-11\zeta{(5)}+6\zeta{(3)}\zeta{(2)}.$$ ...
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3answers
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Why does $\int^{ab}_{a} \frac{1}{x} dx = \int^{b}_{1} \frac{1}{t} dt$?

I can't understand how the integral having limits from $a$ to $ab$ in Step 1 is equivalent to the integral having limits from $1$ to $b$. I'm a beginner here. Please explain in detail. ...
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1answer
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Why does the method to find out log and cube roots work?

To find cube roots of any number with a simple calculator, the following method was given to us by our teacher, which is accurate to atleast one-tenths. 1)Take the number $X$, whose cube root needs ...
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4answers
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How do I cube/square a logarithm?

Btw, please don't give me the answer. I just wanna know how to raise a logarithm to its cube cause I'm stuck in this part, but don't solve it for me. $$\log \sqrt[3]x = \sqrt[3]{\log x}$$ I tried ...
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1answer
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$\log_7 n$ is either an integer or an irrational number

Show that $\log_7 n$ is either an integer or an irrational number where n is a positive number. I assumed that it is rational and tried to get a contradiction for $\log_7 n = a/b$, where b does ...
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2answers
486 views

continuum between linear and logarithmic

A friend and I are working on a heatmap representing some population numbers. Initially we used a linear color scale by default. Then, because the numbers covered a wide range, I tried using a log ...
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3answers
337 views

Solving $\log _2(x-4) + \log _2(x+2) = 4$

Here is how I have worked it out so far: $\log _2(x-4)+\log(x+2)=4$ $\log _2((x-4)(x+2)) = 4$ $(x-4)(x+2)=2^4$ $(x-4)(x+2)=16$ How do I proceed from here? $x^2+2x-8 = 16$ $x^2+2x = 24$ ...
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3answers
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What's a good class of functions for bounding/comparing ratios of complicated logarithms?

I have this goofy series $\sum \limits_{n=2}^\infty \frac{ \log_2 \left[ n \log_2^2 n \right]}{n \log_2^2 n}$ that Wolfram Alpha tells me diverges by the comparison test (and indeed, in the larger ...
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2answers
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Prove that $\log(1 + \sqrt{1+x^2})$ is uniformly continuous?

Problem Prove that $$\log(1 + \sqrt{1+x^2})$$ is uniformly continuous. My idea is to consider $|x - y| < \delta$, then show that $$|\log(1 + \sqrt{1+x^2}) - \log(1 + \sqrt{1+y^2})| = ...
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2answers
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How can I solve $x^x = 5$ for $x$? [duplicate]

Possible Duplicate: Is $x^x=y$ solvable for $x$? I've been playing with this equation for a while now and can't figure out how to isolate $x$. I've gotten to $x \ln x = \ln 5$, which seems ...
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1answer
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How does Knuth's algorithm for calculating logarithm work?

I had a look at Knuth's The Art of Computer Programming, book 1. In chapter 1, section 1.2.2, exercise 25, he presents the following algorithm for calculating logarithm: given $x\in[1,2)$, do the ...
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Logarithm of a Markov Matrix

Start with a Markov matrix $\mathbf{M}$, whose elements are all between $0 \le \mathbf{M}_{ij} \le 1$ and each row sums to one. There is a natural connection with this matrix and the rate matrix ...
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2answers
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Proof $e^x = \exp(x)$?

Define $$\ln (x) = \int^{x}_{1}\frac{1}{t}$$ Assume I have proven that $\ln x$ is one-to-one and therefore has an inverse $\exp (x)$. Define $e$ as: $\ln e = 1$ Now, if you have no other notion ...
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Definition of a logarithm

My question is as follows: Is the below a useful elementary way of dealing with negative arguments? If not, what is a better (elementary or not) way of dealing with negative arguments of the ...
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3answers
209 views

Why is $\int\limits_{1}^{n} \log x \,dx \le \sum\limits_{x = 1}^{n}\log x$?

It has been a long time since I studied integrals, so this question may sound stupid. I was going through this wiki page, and came across the following inequality: $$\int_{1}^{n} \log x \,dx \le ...
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1answer
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Solutions for this logarithmic equation.

For which values of $k$ does the equation $\log_a(kx+3)+\log_a(x+1)=\log_a(2x+1)$ have one or more solutions in $x$? The logarithmic functions must have the restriction that the argument is ...
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2answers
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Is $(\log(n))!$ a polynomially bounded function?

Is the following statement true? How would you prove it? i.e. Is it a polynomially bounded? $$ \lceil \lg(n) \rceil ! \in O(n^k) $$ How about $$ \lceil \lg \lg(n) \rceil ! \in O(n^k) $$ Thanks a ...
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How is $ \sum_{n=1}^{\infty}\left(\psi(\alpha n)-\log(\alpha n)+\frac{1}{2\alpha n}\right)$ when $\alpha$ is great?

Let $\psi := \Gamma'/\Gamma$ denote the digamma function. Could you find, as $\alpha$ tends to $+\infty$, an equivalent term for the following series? $$ \sum_{n=1}^{\infty} \left( \psi (\alpha ...
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2answers
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How to find the limit of $\lim_{n\to \infty} n(H(n) - \ln(n) - \gamma)$

How to find the following limit: $$\lim_{n\to \infty} n(H(n) - \ln(n) - \gamma)$$ where $H(n) = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$ is the $n^{th}$ harmonic number and $\gamma$ is the Euler ...
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3answers
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Computing the value of logarithmic series: $Q(s,n) = \ln(1)^s + \ln(2)^s + \ln(3)^s + \cdots+ \ln(n)^s $

Given a series of the type: $$Q(s,n) = \ln(1)^s + \ln(2)^s + \ln(3)^s + \cdots+ \ln(n)^s $$ How does one evaluate it? Something I noticed was: $$Q(1,n) = \ln(1) + \ln(2) + \ln(3)+ \cdots+\ln(n) = ...
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1answer
263 views

Explain this code to compute $\log(1+x)$

It's well known that you need to take care when writing a function to compute $\log(1+x)$ when $x$ is small. Because of floating point roundoff, $1+x$ may have less precision than $x$, which can ...