Tagged Questions

Questions related to real and complex logarithms.

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1
vote
1answer
68 views

solving 3 unknown with 2 equations in Natural numbers

Iam trying to solve $a^b = f_1$ , $a^c = f_2$, I know the answer will get infinite set of answers. but can I solve it in Natural numbers , hence $a$, $b$, $c$, $f_1$, $f_2$ are natural numbers and ...
1
vote
1answer
42 views

Complexity analysis of logarithms

I have two functions, f(n)=log(base 2)n and g(n)=log(base 10)n. I am trying to decide whether f(n) is O(g(n)), or Ω(g(n)) or Θ(g(n)). I thinks i should take the limit f(n)/g(n) as n goes to infinity, ...
3
votes
3answers
637 views

How to solve: $x+\ln(x^2-1)=0$

I'm trying to solve the following equation for a couple of hours with no success. $x+\ln(x^2-1)=0$ I'm trying to find x. I tried playing with logarithmic identities to transform it to something ...
3
votes
2answers
170 views

Convergence of integral of log and sum the mean of the logs

How can I show that the following limit converges and $L \in (0, +\infty)$? $\lim\limits_{n \to +\infty}\left( S_n - T_n\right)$, where $S_n = \int\limits_1^n \log x\, dx$, and $T_n = \sum_{k = ...
2
votes
4answers
61 views

Arithmetic progression of logarithms

If $\log_{\sqrt{2}}a$, $\log_{\sqrt{2}}(2a^2)$, $\log_{\sqrt{2}}(a^3+4)$ are in A.P, find the value of "$a$". I tried solving this and I am getting $a^3 = \frac{4}{3}$ is it correct. Please ...
1
vote
2answers
179 views

Proof of inequality involving logarithms

How could we show that $$\left|\log\left( \left({1 + \frac{1}{n}}\right)^{n + \frac{1}{2}}\cdot \frac{1}{e}\right)\right| \leq \left|\log\left( \left({1 - \frac{1}{n}}\right)^{n - \frac{1}{2}}\cdot ...
1
vote
1answer
72 views

How to prove: for some $c>0,x>2 , c,x\in \mathbb R , \, \int_2^x \frac{\mathrm dt}{\log t}-\frac{x}{\log x} \leq \frac{cx}{(\log x)^2}$

How to prove: for some $c>0,x>2 , c,x\in \mathbb R$ $$ \int_2^x \frac{\mathrm dt}{\log t}-\frac{x}{\log x} \leq \frac{cx}{(\log x)^2}$$ I have tried my textbook, notes and also tried to find ...
1
vote
2answers
118 views

discrete exponential calculating

I am interesting in about discrete exponential calculating. I know that $a^b = c\mod k$ is calculated as below. for example $3^4 = 13 \mod 17$. $3^4 = 81$; $81 \mod 17 = 13$. I am interesting ...
2
votes
4answers
64 views

Unknown in logarithm base $122312/100000 = (1+t)^5$

I am new to exponents and logarithms, and have been stuck with this for a quite long time. The problem is: $$\frac{122312}{100000}=(1+t)^5$$ or $$\log_{1+t}\frac{122312}{100000}=5$$ I have no ...
1
vote
1answer
86 views

Is $e^{\alpha\log(M)}$ equal to $M^{\alpha}$?

Supposing the matrix logarithm exists, is $e^{\alpha\log(M)}$ equal to $M^{\alpha}?$ This equality obviously holds for positive reals, but does it also hold for matrices?
3
votes
1answer
75 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
531 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
78 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
83 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
207 views

Right angle triangle - Logarithm Problem

Prove that if a and b are the lengths of the legs and c is the length of hypotenuse of a right angle triangle, c-b not equal to 1; c+b not equal to 1 then ...
-3
votes
2answers
307 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
87 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
137 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
302 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
285 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
61 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
84 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
365 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
145 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
703 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
81 views

Why does does 2ln(x) = (ln(x))/5?

According to Google calculator, $2\ln(x) = (\ln(x))/5$ 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
118 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
916 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
274 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
50 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 ...
16
votes
3answers
2k 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
37 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
324 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
71 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
313 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
64 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
111 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
125 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
84 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
45 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
219 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
966 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
56 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
207 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
121 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
103 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
524 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
540 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
700 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
122 views

Two log trig integral

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