Questions related to real and complex logarithms.

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6
votes
9answers
114 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 ...
1
vote
2answers
42 views

Domain of $\log\lvert x^3+1\rvert$

This is a really simple question I think, but I'm looking for justification/clarification as well. I have a function \begin{align} y\left(x\right)=\log\left|x^3+1\right|,\tag{1} \end{align} state this ...
4
votes
5answers
56 views

Why is $\lim_{x \to +\infty}\log x = +\infty$ if $\mathrm{d}/\mathrm{d}x (\log x) = 1/x$?

Why is $\lim_{x \to +\infty} \log(x) = +\infty$? I would have expected that the value of this limit is some fixed number, since $$\frac{\mathrm d}{\mathrm dx} \log x = \frac1x$$ and ...
1
vote
1answer
22 views

Rewriting log of a sum

Suppose we have a vector $$X=[x_1,x_2,\ldots,x_n],\quad x_i\in \mathbb{R} \text{ for } i=1,2,\ldots,n $$ Now if we have a formula $$f_X(x)=\log\left(\sum\limits_{i=1}^nx_i\right)$$ Is it possible to ...
0
votes
0answers
21 views

What are binary operation properties, so sequence fold could be mapped?

I have a sequence of numbers ${x_1, x_2, ..., x_n}$. The task is to compute a value of $S = f(x_1, f(x_2, ..., f(x_{n-1}, x_n)))$ where $f$ is one of: $f(x,y) = log(x \cdot y)$ $f(x,y) = x \cdot ...
7
votes
1answer
89 views

ln(z) as antiderivative of 1/z

When integrating $$\frac{1}{x}$$ (where $x \in \mathbb{R} $) one gets $$ln|x|+c$$ since for $x>0$ $$(ln|x|+c)'=(ln(x)+c)'=\frac{1}{x}$$ and for $x<0$ ...
0
votes
2answers
59 views

Natural Logarithm can't understand properties

~I don't get some of the properties of natural logarithm ($\ln$). $\ln(x^y) = y\ln(x)$ ex. $3\ln 7 = \ln 343$ and what is the difference between the above example and this $3\ln^2(7)$ not equal to ...
1
vote
1answer
10 views

Explaining the non-application of the multiplication law of logarithms, when logs are in the denominators.

I have an A' Levels student who had to solve the following problem: $ log_2 x + log_4 x = 2$ This was to be solved using the Change of base rule, and then substitution, as follows: $ \frac{1}{log_x ...
0
votes
1answer
12 views

Equations transformations with roots

How does the following transformation works (do not write that it is easy i want the answer): $$\ln \sqrt[n]{\frac{n!}{n^n}}=\frac{\ln \frac{n!}{n^n}}{n}$$
2
votes
1answer
46 views

Understanding the proof behind $\pi(x) \ge \frac{\log 2}{2}\frac{x}{\log x}$

I am trying to understand the argument behind the proof that: Given: $$\pi(2n) \ge \log 2\frac{2n}{\log 2n}-1$$ Then for $x \ge 2$: $$\pi(x) \ge \frac{\log 2}{2}\frac{x}{\log x}$$ Here's the ...
25
votes
3answers
840 views

Closed form for $\int_0^\infty\arctan\Bigl(\frac{2\pi}{x-\ln\,x+\ln(\frac\pi2)}\Bigr)\frac{dx}{x+1}$

I'm trying to find a closed form for this integral: $$I=\int_0^\infty\arctan\left(\frac{2\pi}{x-\ln\,x+\ln\left(\frac\pi2\right)}\right)\frac{dx}{x+1}$$ Its approximate numeric value is ...
1
vote
0answers
27 views

what is $\sum_{n=1}^{\infty}\frac{x^{n+1}}{n}-\sum_{n=1}^{\infty}\frac{x^n}{n}$

I posted a question earlier about the taylor of $(1-x)\ln(1-x)$ but i made a miscalculation and decided to delete it, sorry about that. anyways, i solved the miscalculation and i found that ...
0
votes
0answers
29 views

trouble understanding complex logarithms

I am finding the complex logarithm very hard to understand. My text defines $G = \mathbb{C} - \{z \in \mathbb{C} : \Re(z) \leq 0, \Im(z) = 0\}$ and defines the principal logarithm to be the branch of ...
1
vote
0answers
26 views

I'm interested in the solution set satisfying the equation $\log_{10} p\times\log_{10} q=\log_{10} r$

The equation interested in is $\log_{10} p\times\log_{10} q=\log_{10} r$ where $p,q,r\in\mathbb N$ are natural numbers. Here, I want not to consider some trivial solutions that make any one of ...
2
votes
0answers
41 views

Proof of an inequality about primes

I'm very new to number theory and looking for a proof of the following inequality: $$c' \log^{\text{#} \mathbb{P}}{R} \leq \sum \limits_{\substack{n \leq R\\p|n \implies p \in \Bbb P}} 1 \leq c ...
2
votes
1answer
33 views

Asymptotic solution to inequality $x < k \ln(1+x)$

What is an upper-bound on $x$, given that $x < k \ln(1+x)$? I believe that the solution is something of the form $\mathcal{O}(k \ln k)$ but I am unable to prove this. This is my first encounter ...
2
votes
3answers
61 views

Why does $\int\frac{1}{2x+1}dx=\frac{1}{2}\ln|2x+1|+C$?

The way I am thinking is as follows: $$\int\frac{1}{2x+1}\,dx = \int\frac{1}{2}\frac{1}{x+\frac{1}{2}}\,dx = \frac{1}{2}\int\frac{1}{x+\frac{1}{2}}\,dx = \frac{1}{2}\ln\left|x+\frac{1}{2}\right|+C$$ ...
0
votes
1answer
113 views

Interesting problem in logarithms

I know this place isn't for math problems/homework, and believe me I've been trying for a long time to solve this problem (45 mins to 1 hour) and besides I think many would find this useful or at ...
0
votes
2answers
41 views

I'm not great with logarithms so I'd appreciate some help with the following

How is it that $n^{\frac{1}{\log n}} = 10$. I understand that $10^{\log a} = a$ but I don't know how to make the correct algebraic manipulations. Note: Assume $log$ is base 10
1
vote
1answer
47 views

Forward Algorithm Hidden Markov Model matrix help [Discrete]!

So this may seem like a bioinformatics question but it is the math part that is giving me trouble. I'm using a Python package called YAHMM to model DNA sequences. I created a model with two states ...
0
votes
2answers
31 views

Is a dollar gain truly equal to a dollar lost?

Under the context of the asymmetrical nature of gain to a loss, as shown below, is 1 dollar gained truly equal to 1 dollar lost? If not how would you go about calculating the equalization ratio at ...
2
votes
3answers
43 views

Integral of $\frac{1}{2x}$

The integral of $\frac{1}{2x}$ is $\frac{\ln(x)}{2}$, but can't it also be $\frac{\ln(2x)}{2}$ or $\frac{\ln(3x)}{2}$? Is there a special reason for $\ln(Ax)$ to have identical derivatives?
8
votes
6answers
178 views

Why is $\ln(x^x)=x\ln(x)$ valid?

I know that $\ln(x^k)=k\ln(x)$ for any constant $k$, but why is $\ln(x^x)=x\ln(x)$. The exponent $x$ is not constant.
2
votes
4answers
65 views

How to solve $4x-\log(x) = 0$

I have a problem solving this equation: $4x-\log(x) = 0$. I can't seem to get this equation to a simpler form featuring $\log$s only or getting rid of the $\log$. Is there a way to solve it without ...
0
votes
1answer
26 views

Best fit in logarithmic chart

I have several variances ($\sigma^2$) which value depends on the velocity ($v$). As you can see in the graph, if increase the velocity, the variance does the same. I am studying this dependency, but ...
2
votes
2answers
27 views

Evaluating Logarithmic Expressions

Evaluate: $$\log_4 \left(\dfrac{1}{256}\right)$$ I am not sure how to approach this since there is nothing set equal to it.
3
votes
1answer
91 views

Integration of (Tsiolkovsky) rocket equation

The (Tsiolkovsky) rocket equation states that the velocity of a rocket can be calculated as $$v(t) = v_0 \ln\frac{m_0}{m_0 - \dot m t}$$ where $m_0$ is the starting mass, $\dot m$ is the (constant) ...
0
votes
1answer
16 views

How to normalize data in another scale?

Let $A$ be a set of values $\{a_1,a_2,a_3,a_4,a_5\}$ where $a_1 = 2$, $a_2 = 1$, $a_3 = 4$, $a_4 = 1$ and $a_5 = 2$, so, the $avg(A) = 2$. I'm looking for a normalization where the values below the ...
2
votes
0answers
36 views

Finding how many solutions does $f(x)=\ln x-kx$ has for $k>\frac 1 e$ and logarithmic inequality question

Find how many solutions does $f(x)=\ln x-kx$ has for $k>\frac 1 e$. $f<0$ at $x\to \infty$ and $x\to 0$. The derivative has a solution only at $x=\frac 1 k$. So place that point in $f$ and ...
0
votes
3answers
54 views

Simplifying Quadratic Equations In Logarithmic Form

$log_{10}(x^2-x-7)=0.1$ $log_{10}(x-8)=1-log_{10}(x+1)$ $log_{10}(x+9)=1+log_{10}(x+1)-log_{10}(x-2)$ Note: I solved them as follows: $x = 3, -2$ but the textbook i'm using said there was no ...
0
votes
0answers
16 views

How to steepen logarithmic function without reducing constant of deceleration

As you can see I have plotted my points in Geogebra and compared them to the function $ y=log_{10}x $ They clearly don't coincide, how would I go about adjusting the function in order to find the ...
3
votes
1answer
30 views

Prove convergence of this generalized integral

Prove the convergence of $$\int_0^1 \left[\ln\left(1+\frac1x\right)\right]^a\mathrm dx$$ for $ a>0$.
0
votes
2answers
76 views

Compute $\lim\limits_{n\to \infty} \ln(3n+7) - \ln(n)$

The reason why I'm having trouble with this problem is because it involves natural log (ln) and I need to find the limit. I need to find $\lim_{n\to\infty} \ln(3n+7)-\ln(n)$. I noticed that as $n$ ...
1
vote
2answers
55 views

Solve the logarithmic equation by $x$

Solve the eqation for all real $x$: $\log_2(x^2+7)+\log_3(x+6)=6$. What I tried: $\log_2(x^2+7)=a$ and $\log_3(x+6)=b$, then $a+b=6$ and $2^a=3^{2b}-4\cdot3^{b+1}+43$. But the problem is $a$ and $b$ ...
0
votes
2answers
90 views

Solving for n, n is an exponent.

If you have a sequence of random numbers ranging between 1 and 64, what is the length of a sequence that will give a 98% chance of having at least one ( 1, 2, or 3) in the sequence? Here is ...
1
vote
1answer
21 views

Multi-variable calculus involving $\ln$

I am having difficulty with differentiating this equation with respect to $y$: $$ W= x^{y \ln(z)}. $$ Differentiating calculators are giving me the answer $$\ln(x) \ln(z).x^{y \ln(z)}$$ But I ...
0
votes
1answer
33 views

Is it true that $x^n <\epsilon \Rightarrow n < \frac{\ln \epsilon}{\ln x}$?

Let: $0 \lt x \lt 1$ $\epsilon > 0$ I need to show that there exists an $N(\epsilon,x)$ such that: $n\ge N(\epsilon,x) \Rightarrow x^n < \epsilon$ This is what I've tried: $x^n ...
1
vote
1answer
30 views

How to “see” that this expression is $>0$.

$N \in \mathbb N$. $\displaystyle\int_{N-1}^N \left(\dfrac{1}{x} - \dfrac{1}{N}\right) dx>0$ This is the finish of a proof, a modification of $\log N-\log (N-1) -\frac{1}{N}$. Calculating it ...
4
votes
2answers
48 views

If $\log_{12}54=a$ then $ \log_{6}12=?$

I am given $$\log_{12}54=a$$ So what will be value of $ \log_{6}12?$ I used base changing theorem and wrote expression as $$\frac{\log_{6}54}{ \log_{6}12} =a$$ And then $$ \frac{1+\log_{6}9}{ a} = ...
0
votes
1answer
33 views

How to solve equations containing logarithms and exponentials

Equation 1: $x+e=e^x$ According to Wolfram alpha : Solution of x $\approx$ -2.6 and 1.4 Equation 2: $x-e = \ln(x)$ According to wolfram alpha, Solution for x $\approx$ 0.07 and 4.1 How does ...
0
votes
1answer
18 views

Properties of Geometric Series

If we have a geometric series $(x_1, x_2, ..., x_{n-1}, x_{n})$ of reason $q$, we can determine the general term formula to be: $x_{1}q^{n-1} = x_{n}$ But by taking the logarithm of the equation we ...
0
votes
1answer
12 views

Linking summations with their correct function(s)

Guys can you please guide me step by step on how to link given functions with the functions to choose from. So for example a function $g(n)\in \Theta n^2$ and if there is no match then you say there ...
1
vote
4answers
63 views

If $a^x=b$, then $ x=$?

Stupid question, I know, but I couldn't remember nor find information by googling on how to find the exponent of $a$ that gives $b$ as the result. If $a^x=b$, then $x=log_a b$ but how do you find $x$? ...
1
vote
4answers
83 views

How to determine the monthly interest rate from an annual interest rate

I have a calculation which gives me the annual interest rate if I already know the monthly interest rate as follows: (Monthly interest rate + 1)^12 In this case I ...
2
votes
2answers
45 views

Solving inequality involving square root and division by logarithm.

I would like to solve the inequality $\sqrt n<\frac{n}{\log(n)}-2$. for some reason I had never done this before. This is clearly the same as $\frac{n}{\log(n)}-\frac{n}{\sqrt{n}}>2$. Which is ...
1
vote
0answers
25 views

Homework : Anti log expression

I have this expression $x(r) = y(a)r^a$ where $r$ is a random variable and I want to express the expression in terms of $r$. The objective is to substitute the variable $r$ into the pdf of $r$, ...
1
vote
1answer
45 views

Solving natural logarithms with absolute value

Question from my text: $e^{4x-2014} - 7 = |-3|$. I've never seen this before and my text is useless! Thank you!
1
vote
0answers
22 views

Notation question

What exactly do you think $\ln^rn$ means in this context? "Prove that the relation $\tau(n) = O(\ln^rn)$ is false for all fixed powers $r$." Where $\tau$ is the divisor function.
0
votes
1answer
18 views

Log inequality- is $(\lceil\log x\rceil - \lfloor\log m\rfloor)\cdot m+2^{\lfloor\log m\rfloor+1}\leq m\cdot(\lceil\log\frac{x}{m}\rceil+2)$?

I'm having some hard times making a tight analysis of the memory requirements for my algorithm. I want to show the following inequality, which will show my data structure can use about 2 bits per ...
2
votes
2answers
47 views

Log inequality - is $\lceil\log x\rceil - \lfloor\log y\rfloor\leq \lceil\log\frac{x}{y}\rceil+1$

Is it true that $$\forall x>y\in\mathbb N:\lceil\log_2 x\rceil - \left\lfloor\log_2 y\right\rfloor\leq \left\lceil\log_2\frac{x}{y}\right\rceil+1$$? I reached this inequality when further ...