Questions related to real and complex logarithms.

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3
votes
1answer
78 views

Solving the recurrence: $h(i) = h\left(\left\lfloor\frac{i+1}{d}\right\rfloor\right)+1 $

I want to solve the following recurrence: \begin{equation} h(1) = 0\\ h(i) = h\left(\left\lfloor\frac{i+1}{d}\right\rfloor\right)+1 \end{equation} What are some basic "methods" I can use to guess a ...
2
votes
1answer
635 views

Rearranging a logarithmic equation

I'm building a web app that displays the frequency of a sound. I have an equation that returns a pixel that a particular frequency should be mapped at. However I would like to reverse the equation so ...
1
vote
3answers
87 views

basic rules logarithm of exponential

I am looking for proof of the basic rules of logarithm. I can prove all basic rules except this $$\log_ab^y=y\log_ab$$ how to get this rule using definition of logarithm.
2
votes
3answers
87 views

Powers and the logarithm

By example: $4^{\log_2(n)}$ evaluates to $n^2$ $2^{\log_2(n)}$ evaluates to $n$ What is the rule behind this?
0
votes
1answer
286 views

Right angle triangle - Logarithm Problem

Prove that if $a$ and $b$ are the lengths of the legs and $c$ is the length of the hypotenuse of a right angle triangle, $c-b \neq 1$, $c+b \neq 1$ then ...
-3
votes
2answers
319 views

Proving Logarithm by substitution

Prove that if $\alpha = \log_{12}18$ and $ \beta = \log_{24}54$ then $ \alpha \beta +5(\alpha - \beta)=1$
0
votes
3answers
90 views

Solving the following equation: $2^{x}+\log_{10}x-2=0$

How can I solve the following equation: $$2^{x}+\log_{10}x-2=0$$ Any help welcome. Thanks!
1
vote
0answers
161 views

Is this “Elegant” algorithm for logarithm by Zeckendorf representation, the same as an 'efficient' algorithm?

The algorithm here which computes the exponent $b$ given a base $a$, and given $n$ = $a$^$b$, appears no better to me than simply counting the number of times we divide $n$ through by the base $a$ ...
1
vote
2answers
353 views

Inverse of natural log

I have a problem: Let $f(x)=\ln(x)$ solve each of the following equations for $x$. the question is in three parts $(f(x))^{-1}=5$ $f^{-1}(x)=5$ $f(x^{-1})=5$ My understanding is that $\ln(x)$ is ...
2
votes
3answers
327 views

Complex Logarithms: Detailed explanation for why $\operatorname{Log} z^2$ is not equal to $2\operatorname{Log}z$

Why is $\operatorname{Log} z^2$ not equal to $2\operatorname{Log} z$ where $z$ is a complex number. $\operatorname{Log} z$ here refers to just the principal Log. Detailed explanation would be ...
1
vote
1answer
70 views

Finding a tangent using a point that is undefined for the function

$f(x) = x\ln(a^2x^2), a > 0$ A tangent to the derivative of the function goes through $(0, 0)$. The task is find the tangent's intersection point with the derivative and the function of the ...
3
votes
2answers
87 views

Calculator question involving $\log_2$?

I have a question, I have a calculator that does $\log$ but I think it does it it in a base ten format for example $\log_{10}(100)=2$ I am wondering how I can solve $\log$ using a base of 2 for ...
2
votes
1answer
402 views

Big o notation $( n \log n + n \log(n^{\log n}))$

I'm trying to transform this: $$n \log n + n \log(n^{\log n})$$ into big O notation. I can't get to reduce the right part of the addition... Neither of these work: $$n^{\log n} ...
3
votes
1answer
158 views

how to test whether an unknown function is logarithmic function or not? can we figure that out by “ the fundamental property of a logarithm”

My question is derived another question:How do we know that ln function (natural log) is a logarithmic function if we start from its calculus definition? I recently learned the calculus definition of ...
1
vote
2answers
969 views

Natural logarithm limit

Is $$\lim_{n\rightarrow +\infty}\ln\left(\frac{n+1}{n}\right)=0?$$ Because it is $\ln(1+\frac{1}{n})$ and $\frac{1}{n}$ tends to $0$, since $n$ tends to infinity, so the limit becomes ...
0
votes
2answers
97 views

Why does does $2\ln(x) = \frac{\ln(x)}{5}$?

According to Google calculator, $2\ln(x) = \frac{\ln(x)}{5}$ for many values of $x$. As I remember my logarithm rules, I don't understand why this should be. Can anyone explain?
1
vote
0answers
121 views

Complex Logarithm

For what values of $p$ is the following valid? $$\log(z^p) = p\log(z)$$ where $$\log(z) = \ln(|z|) + i[\arg(z)+2\pi n]$$ where $n$ is an integer. I heard the expression above should not be valid for ...
3
votes
2answers
1k views

Integral of Natural Logs

I had this problem on an integral test today. I tried using u substitution but to no avail. Integral: $\int (1+\ln(x))x\cdot \ln(x)dx$.
1
vote
2answers
343 views

Express the following formula in terms of n

Express $$ T(2^k)=\frac{k(k+1)}{2}. $$ In terms of $n$, where $n = 2^k$. I'm not sure how to go about with the conversion. Can someone concisely explain? Thank you.
-2
votes
1answer
55 views

Lower bound for $\ln x$ using Lagrange's mean value theorem or Rolle's theorem

I have to prove this inequality. $$ \ln x>\frac{2(x-1)}{x+1} \hspace{15pt}, \hspace{15pt}\text{where}\hspace{5pt}x>1 $$ using either Lagrange's mean value theorem or Rolle's theorem. Can ...
17
votes
3answers
4k views

When log is written without a base, is the equation normally referring to log base 10 or natural log?

For example, this question presents the equation $$\omega(n) < \frac{\log n}{\log \log n} + 1.4573 \frac{\log n}{(\log \log n)^{2}},$$ but I'm not entirely sure if this is referring to log base ...
2
votes
0answers
38 views

Compute $ \operatorname{Li}_{3}\left(\frac{1}{2} \right) $

Where could I find a proof of the identity $$ \operatorname{Li}_{3}\left(\frac{1}{2} \right) = \sum_{n=1}^{\infty}\frac{1}{2^n n^3}= \frac{1}{24} \left( 21\zeta(3)+4\ln^3 (2)-2\pi^2 \ln2\right)$$ ?
0
votes
2answers
347 views

basic math question: transform a sum of exponents to a sum of logarithms

I am sure this is a really dumb question but I am having trouble understanding it since I do not have any math background. I have the logarithms of 2 values: ...
2
votes
1answer
80 views

Logarithm problem

I have an easy problem. I can see the answer but I don't know how to solve the problem and get the answer "the mathematical way". The statement is: $x \large {\cdot 2^{\log _x 5 } = 10 }$ Then I ...
1
vote
2answers
399 views

Find the limit of the sequence containing logarithm??

Find $\lim_{n→∞} [log(2+3^n)]/2n$ I have my work till the very last step then i dont know how to continue $\lim_{n→∞} [log(2+3^n)]/2n$ =$\lim_{n→∞} log(3^n)+\lim_{n→∞} log[(2+3^n)/3n]$ ...
1
vote
1answer
71 views

On the uniqueness of the real logarithm of a real matrix

I was wondering about the uniqueness claim in the paper, on the exitence and uniqueness of the real logarithm of a matrix, to answer the questions but I have not been able to understand the ...
1
vote
1answer
129 views

A simple inequality with logarithms and exponential

I want to prove that for $k>0$: $ 2^k \geq \frac{-1}{\log_2(1-\frac{1}{2^k})}$ I've plotted both functions and it seems to be the case for k>0. In fact, it would also be nice to see that: $ ...
0
votes
4answers
211 views

How to understand that sequence is logarithmic?

Let's say I have example of phonebook lookup. I need to find one record in it. I can always divide phonebook into 2 equal parts and try to find a record in that way. ...
-1
votes
1answer
93 views

Find a relation between and y that does not involve logarithms

Could I please have a solution to this, I've spent an hour on it so far -_- Thanks in advance. $$ \log_{10}(1+y) - \log_{10}( 1-y) = x$$
1
vote
0answers
48 views

$f(n) = n^2 \lceil \log n \rceil$ is time constructible

I have a question, I want to show, that: $$f(n) = n^2 \lceil \log n \rceil $$ is time-constructible. I have no idea how to do this. I know that $n^2$ is time-constructible and I know that $\log n$ ...
4
votes
2answers
257 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 ...
1
vote
1answer
1k views

Another two hard integrals

Evaluate : $$\begin{align} & \int_{0}^{\frac{\pi }{2}}{\frac{{{\ln }^{2}}\left( 2\cos x \right)}{{{\ln }^{2}}\left( 2\cos x \right)+{{x}^{2}}}}\text{d}x \\ & \int_{0}^{1}{\frac{\arctan ...
0
votes
2answers
58 views

For what $f(n)$ does $O(f(n) \log n)=O(\log\log n)$?

$k=f(n)$. Given $O(k \log_2 n)$, what function $f$ of $n$ would be needed for it to equal $O(\log_2 \log_2 n)$? (where $k \in n \in \mathbb{Z}^+$)
2
votes
4answers
220 views

Solving $\;5^{2x}-4\cdot 5^x=12$

I need to solve $\quad\displaystyle 5^{2x}-4\cdot 5^x=12$. I've only gotten this far: $\quad \displaystyle 5^{2x}-20^x=12.$ I don't know what to do next. Thanks in advance!
2
votes
2answers
130 views

About the use of Stirling approximation

How to prove this inequality: $$\ln \Gamma \left( x \right)-2\ln \Gamma \left( \frac{x+1}{2} \right)>\frac{2x}{3}$$ Sry I forgot to mention that $x>300$
2
votes
1answer
106 views

Some log and exponential integral

I think these should hav some closed form: $$\displaystyle\begin{align*} & \int_{0}^{1}{\frac{\left( 1-x \right)\ln \left( x \right){{\text{e}}^{-x}}}{\pi -x}}\text{d}x \\ & ...
3
votes
3answers
798 views

Solving inequality involving logarithms

I must be doing something wrong. I want to solve the following, where n is a positive integer, and p is a real number between 0 and 1. $$(1-p)^n \le 0.4$$ So I take the log on both sides: ...
2
votes
5answers
602 views

Convergence of series $\sum_{n=1}^\infty \ln\left(\frac{2n+7}{2n+1}\right)$?

I have the series $$\sum\limits_{n=1}^\infty \ln\left(\frac{2n+7}{2n+1}\right)$$ I'm trying to find if the sequence converges and if so, find its sum. I have done the ratio and root test but It ...
1
vote
2answers
912 views

Log laws and modulus

If you have the log of a modulus, (like after integration), how do the log laws work? So if you have $a\ln\left|2x-3\right|$ does it become: $\ln\left|(2x-3)^a\right|$ or $\ln(\left|2x-3\right|)^a$, ...
2
votes
1answer
132 views

Two log trig integral

$$\begin{align*} & \int_{0}^{\frac{\pi }{2}}{{{\ln }^{n}}\sin x\text{d}x} \\ & \int_{\frac{\pi }{4}}^{\frac{\pi }{2}}{\ln \left( \ln \tan x \right)}\text{d}x \\ \end{align*}$$
3
votes
3answers
200 views

How can I show that $f(x) = (x^2)/(1-e^x)$ has global minimum at $(0, +\infty)$?

I showed that $\lim f(x) = 0$ at both the $0$ end and $+\infty$ end. What is the proper way to finish the proof?
0
votes
0answers
60 views

What does taking the logarithm of a variable mean?

Question with regards to taking the logarithm of a variable (Statistics Question) Say you have a bar graph displaying data for an example "Cost of Computer Orders by the Population" and you are ...
0
votes
1answer
84 views

Making data fit a certain equation form?

I have a set of data that follows a polynomial (parabola) equation almost perfectly, but I need it to be of the form $T(n) = a\cdot n + b\log(n)$ I approximated it to the polynomial equation using ...
4
votes
1answer
170 views

Solve $8 \log(x) - x = 0$

Someone came to me recently with this seemingly simple equation to solve: $$8 \log(x) - x = 0$$ So far, everything I have tried has been a dead end. Is there a symbolic solution to this kind of ...
1
vote
1answer
93 views

$x \sim y \implies \pi(x) \sim \pi(y) $ and repeated applications of PNT

Let $\sim$ mean if $a \sim b$ then $\lim_{x \to \infty} \frac{a}{b} =1.$ The following is a threshold question. It seems that $x \sim y \implies \pi(x) \sim \pi(y).$ Pf. $\pi(x) \sim \frac{x}{\log ...
3
votes
3answers
162 views

$x \sim y \implies \log x \sim \log y$?

Does $x \sim y \implies \ln x \sim \ln y$? I would have thought not, but the following has almost persuaded me otherwise: Assume $x \sim y.$ Does this imply that $$\tag{1}I = ...
3
votes
1answer
313 views

exponential population growth models using $e$?

Im trying to understand this write up [1] of cell population growth models and am confused about the use of natural logarithms. If cells double at a constant rate starting from 1 cell, then their cell ...
5
votes
3answers
222 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 ...
1
vote
1answer
279 views

log-odds to probabilities

For example the variance in the log-odds is $0.07$ $(0.01)$, and the mean log-odds is $-0.65$ $(0.03)$. Just transforming this variance to probability given me $0.51$. So if the probabilities have a ...
1
vote
0answers
50 views

Calculating the number of states in power of 10's

This is part of a problem that was assigned to me. It might seem elementary, but I need a hint on how to start this problem. I have 10 billion bits that can either be on or off at any time. ...