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Questions tagged [logarithms]

Questions related to real and complex logarithms.

2
votes
0answers
35 views

Prove $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}\ln{(2n+1)}}{2n+1}=\pi/4(\gamma-\ln{\pi})+\pi\ln{(\Gamma(3/4))}$

In the title, $\gamma$ is the Euler-Mascheroni constant and $\Gamma(3/4)$ represents the extension of the factorial function. This isn't a homework question or something, someone left it on a board ...
0
votes
2answers
45 views

Closed form for $\sum_{n=1}^\infty \log(n) * x^n$

As in the title, I'm in quest for $\sum_{n=1}^\infty \log(n)\cdot x^n$, where $0 \le x \lt 1$ Wolfram Alpha says: $-\operatorname{PolyLog}^{(1, 0)}(0, x)$, but I don't understand what that means. (...
-2
votes
2answers
36 views

Find the S value - Logarithmic summation [on hold]

Find the value $S$. $$ S = \frac{1}{\log_2 1000!} + \frac{1}{\log_3 1000!}+\frac{1}{\log_4 1000!}+...+\frac{1}{\log_{999} 1000!} + \frac{1}{\log_{1000} 1000!} $$ Any ideas?
-1
votes
2answers
17 views

Simplify $\log(α+ b(x- t)+ k)$

I am trying to simplify this natural log expression the best that I can, but I am unsure what to do in order to separate $b(x-t)$. Would it be $\log(b) + \log (x/t)$? so would the whole thing be: $$ ...
0
votes
0answers
34 views

Studying the continuity of the function ln and exponential : [on hold]

I didn't find the right way to find if ln is continuous starting from the fact that exponential is continuous "knowing that they are bijective".
0
votes
4answers
60 views

Show that $x \ln(x)-1=0$ at least has one solution on $\Bbb R$

Show that this equation at least has one solution on $\Bbb R$ (Find the solution of this equation on $\Bbb R$) $$x\ln(x)-1=0$$ We know that : ( $\operatorname{Log}_e(a) = \ln (a) $ ) Thanks in ...
6
votes
3answers
138 views

Solving the equation $\frac{\ln (x)}{\ln (1-x)} = \frac{1}{x} - 1$

I'm trying to solve the following equation (which has solution $x = 1/2$) $$\frac{\ln (x)}{\ln (1-x)} = \frac{1}{x} - 1 $$ I can't seem to do it analytically. Any ideas?
1
vote
1answer
20 views

Explanation of summation equation

Could somebody please explain the following equation to me? I have no clue what H represents, nor how theta(ln(n)) - theta(ln(k)) results in theta(ln (n/k)) Any explanation would be appreciated. ...
0
votes
5answers
56 views

Find the solution to $x^{(\log_5 x^2 + \log _5 x-12)}=\frac{1}{x^4}$

Find the solution to $x^{(\log_5 x^2 + \log _5 x-12)}=\frac{1}{x^4}$ I equated their exponents, That gave me $\log_5 x = \frac{8}{3}$ But the answer given in my book is $1$. Obviously, 1 satisfies ...
0
votes
0answers
15 views

Creating a logarithmic function from a graph (Half-Time) [on hold]

I have a set of X values and their corresponding Y values, I completely understand the values, I know they're a logarithmic function, but I don't know how to graph it. For X = 0, 1, 2, 3, 4, 5 Y = 0, ...
2
votes
3answers
48 views

Show that $\lim_{n\to\infty}\frac{\log_an}{n} = 0$ for $0<a<1$

Let $0<a<1$, prove that: $$ \lim_{n\to\infty}\frac{\log_an}{n} = 0 $$ I've started with proving a simpler case for $a>1$. Choose some $\varepsilon >0$ such that: $$ \frac{\log_an}{n} &...
3
votes
2answers
55 views

Limit, Riemann Sum, Integration, Natural logarithm

For any natural number $m$, $\lim_{n\rightarrow \infty }\left ( \frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+\cdots +\frac{1}{mn} \right )=\ln (m)$. I tried to prove the statement in the following way. ...
0
votes
2answers
27 views

Use the properties of logarithms to write the expression as a single term

I am asked to use the properties of logarithms to write the following expression as a single term: $(1/2)\ln(4t^4) - \ln b $ I have the solution here but I get stumped halfway through: $(1/2)\ln(4t^...
3
votes
1answer
59 views

Asymptotic behavior of roots of an equation involving exponential and logarithm

Prelude This Post is a continuation of this Original Post. The original problem asked is: How many solutions does the following equation have: $$ a^x = \log_a(x) \,,\quad a \in (0,1) \wedge x \in\...
2
votes
4answers
59 views

Comparing $\ln 1000$, $\sqrt[5]{1000}$, $3^{1000}$, and $1000^{15}$ without calculator

In my Pre-Calculus class we were given the following problem: Put the following four values in order from smallest to largest: $\ln 1000$, principal $5$th root of $1000$, $3^{1000}$, and $1000^{15}$...
0
votes
1answer
17 views

How to solve and find the value of $\log k$ equation

I am doing a mathematics problem where the problem equation led me to below conclusion: $$\log k = 52.79$$ Now I am not sure how to solve and get value of $k$? What way we can find the value of $k$?
5
votes
4answers
95 views

A nice relationship between $\zeta$, $\pi$ and $e$

I just happened to see this equation today, any suggestions on how to prove it? $$\sum_{n=1}^\infty{\frac{\zeta(2n)}{n(2n+1)4^n}}=\log{\frac{\pi}{e}}$$
3
votes
1answer
43 views

Combinatorial Proof that the Logarithm of a Product is the Sum of the Logarithms

I've been strongly drawn recently to the matter of the fundamental definition of the exponential function, & how it connects with its properties such as the exponential of a sum being the product ...
2
votes
1answer
60 views

Is there a function that satisfies $f(\log \frac{p}{1-p}) = p(1-p)$?

Let $p\in (0,1)$ I am wondering there exists a function $f:\mathbb{R}\to[0,{1\over 2}]$ that satisfies $$ f\left(\log \frac{p}{1-p}\right) = p(1-p) $$ I've tried messing around with exponentials ...
6
votes
2answers
112 views

How many solutions are there for the equation $a^x = \log_a x$, where $0 < a < 1$?

How many solutions are there for the equation $a^x = \log_a x$, where $0 < a < 1$? When I first saw this quiz for japanese high school students, I wondered there was only 1 solution for the ...
4
votes
1answer
82 views

Evaluate $\int \frac{\ln(t+\sqrt{t^2+1)}}{1+t^2} \, dt$

$$I=\int \frac{\ln(t+\sqrt{t^2+1)}}{1+t^2} \, dt$$ i used substitution $t=\tan y$ so $$I=\int \ln(\sec y+\tan y)dy$$ Using integration by parts we get: $$I=y \ln(\sec y+\tan y)-\int y \sec ydy$$ ...
-3
votes
0answers
50 views

why $\lim \limits_{x\to \:-\infty \:}\left(\ln \left(x\right)\right)=\infty \:$ [closed]

If the domain of $\ln$ is $(0,\infty)\;$ and $\lim \limits_{x\to \:-n \:}\left(\ln \left(x\right)\right)=DNE \:$, $n \in \mathbb{N}$ then why $\lim _{x\to \:-\infty \:}\left(\ln \left(x\right)\right)=...
1
vote
1answer
35 views

Addition of Logarithmic Equation

According to the basic rules of $\log$, I'm solving both $\log$ terms as for first: base is $3$, $N$ is $9$ so exponent is calculated as $2$, and same for other term. But I'm confused with this '$x$'. ...
3
votes
4answers
42 views

The product of the two roots of $\sqrt{2014}x^{\log_{2014} x}=x^{2014}$ is an integer. Find its units digit

The product of the two roots of $\sqrt{2014}x^{\log_{2014} x}=x^{2014}$ is an integer. Find its units digit. I'm quite unable to solve the problem given. I have no idea how to deal with that $\sqrt{...
-5
votes
2answers
29 views

Solving exponential equations 1 [closed]

there is a algebric quation given below. 2^(x)+4^(y)=2^(y)+4^(x) is solved for y in terms of x` ; where x<0, then the sum of solution =?
0
votes
3answers
38 views

Why is it that $\ln\Big(1+\frac{x}{y}\Big)=\ln(1+\exp(\ln x - \ln y))$?

Is the relation $\ln\Big(1+\frac{x}{y}\Big)=\ln(1+\exp(\ln x - \ln y))$ an approximation? If so, how can I derive this relation?
0
votes
2answers
21 views

Prove that two functions grow at equal rate

Prove that functions $g(x)=\ln(\ln(x))$ and $h(x)=\ln(\lg(x))$ grow at equal rate for every base and value of x. I'm actually very confused about what 'for every base' actually means. I'm assuming ...
0
votes
1answer
22 views

Radius of convergence for $\ln(a+x)$

Since the radius of convergence $R$ for the Taylor series of $\ln(z)$ around $1$ is $1$, i.e. $$ R\left\{\ln\left(1+z\right) = \sum \frac{(-1)^{k-1}}{k}x^k \right\} = 1 ,$$ does this mean that for $...
0
votes
3answers
32 views

Find whether $f(x)$ is $O(g(x))$ [whether $f(x)$ is Big-O of $g(x)$]

Given: $f(x)=3^{\sqrt{x}}, g(x) = 2^x$, find whether $f(x)$ is Big-O of $g(x)$, and vice-versa. I want to use the following fact: $$\lim_{x\to\infty}(\ln|f(x)|-\ln|g(x)|) \leq ln(C) \implies f(x)=O[g(...
0
votes
2answers
46 views

Given: Log2=a, Log7=b. Find: Log 56.

I don't know how to solve this. Can someone help me? How do I use the information above to help me find Log 56?
1
vote
2answers
44 views

Finding derivative of $f(x)$ where $f(xy) = f(x) + f(y)$ - without change of variable

Let $f(x)$ be a function $(0,\infty) \to R$ and for every $x,y$ in the domain we have: $$f(xy) = f(x) + f(y)$$ It is like logarithm but we don't know the exact form of the function. we know it is ...
1
vote
1answer
43 views

Use composition of functions to prove that $f(x) =e^x-e^{-x}$ and $g(x) = \ln\left(\frac {x+\sqrt{x^2 + 4}}{2}\right)$ are inverses [closed]

I have found no way of composing these two functions $$f(x) =e^x-e^{-x} \quad g(x) = \ln\left(\frac {x+\sqrt{x^2 + 4}}{2}\right)$$ that has proven these two are inverses. If there is anyone that ...
0
votes
2answers
19 views

Logarithms in Summations : Confusion!

I see this simplification and I am confused! I thought there is no explicit way to simplify the logarithm of a summation. Can someone explain how the the second term( involving the summation), gets ...
1
vote
1answer
36 views

$\lim_{r\to\infty}\int_{c-ir}^{c+ir}z^{-\lambda}(\log(z))^{-1}dz$

Let $c>1$ and $\lambda\in\mathbb{C}$ with $\Re(\lambda)\geq1 $. Show that $\lim_{r\to\infty}\int_{c-ir}^{c+ir}z^{-\lambda}(\log(z))^{-1}dz=0$, where the path of integration is the straight line and ...
0
votes
1answer
31 views

Proof of log identity for positive definite matrices

On Wikipedia, it is claimed that $\rm{tr}(\log(AB)) = \rm{tr}\log(A) + \rm{tr}\log(B)$. The result is valid only if $A$ and $B$ are positive definite. I use this result in entropy calculations but ...
2
votes
2answers
23 views

How to show that the natural logarithm is Lipschitz on $[\beta, \infty)$

I want to show the following result: Let $\ln(x)$ have domain $D = [\beta, \infty)$ then $|\ln(x) - \ln(y)| \leq \dfrac{1}{\beta} |x-y|, \forall x,y \in D$ I am confused as to how to prove this ...
0
votes
1answer
35 views

Prove that $\mathrm{arcsinh}(x)$ is an odd function

The inverse hyperbolic sine $\sin^{-1}(x) = \mathrm{arcsinh}(x)$ is an odd function. This can be proved by manipulating the expression $\mathrm{arcsinh}(-x) = y$ as shown here. But how to prove it ...
2
votes
2answers
113 views

Why are we using “Euler's Number” constantly? [duplicate]

I am a student at technical university.We are currently studying Calculus and I am really curious about why we are using Euler's Number.
1
vote
0answers
35 views

Finding the limit of $\frac{x}{e^x}$ from the graph of reciprocal function

The problem asks (teachers) to suggest an activity of three questions to students based on this picture to show that $\lim\limits_{x \to +\infty} \dfrac{x}{e^x}$ (picture : part of the hyperbola ...
1
vote
1answer
30 views

Exponential Approximation

I was recently reading a text book on random walk. In the proof of a local central limit theorem the book used the following step: $e^{-2k^2/(n+k)} = e^{-2k^2/n} \space exp\{\frac{2k^3}{n^2} +O(\frac{...
1
vote
1answer
35 views

Solving sets of 3 equations with 3 unknown with some logarithms thrown in.

I'm having some difficulty solving the following set of equations: $6.16=x\log{(100y+z)}\\ 8.59=x\log{(250y+z)}\\ 12.72=x\log{(1000y+z)}$ The sum of the two different variables in the logarithm is ...
0
votes
0answers
16 views

numerical analysis and weakness of calculator

Suppose that our calculator can calculate $e^x$ so well.But its program for calculating $\ln (x)$ is poor. How to improve the accuracy of $ln(x)$ by using the fact that $$ln(a)=b+ln(1+\frac{a-e^b}{e^...
2
votes
3answers
85 views

Is $1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{x} < \log_2 x$

If $x \ge 5$, is $1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{x} < \log_2 x$ I believe the answer is yes. Here is my thinking: (1) $\log_2{5} > 2.32 > 2.284 > 1 + \frac{1}{2} + \...
0
votes
0answers
50 views

What is the value of following limit?

Let $P$ be a polynomial in complex variable $z$ of degree $d$ i.e. $P(z)= a_d z^d+.....+a_1 z+a_0$ Now I want to calculate following limit $f(z) = \limsup_{n \to \infty} \frac{1}{d^n} (Log|P(z)^{*...
0
votes
0answers
19 views

How to take log of sum product of matrix and e^x?

I wonder how to take log of this sum function where it has $e$ inside $$f(x) = \sum_{ij}V_{ij}\exp\left((x-N_{ij})^2\right)$$ where $V_{ij}$ and $N_{ij}$ are matrix size $i \times j$
-1
votes
1answer
23 views

Simple inequality with natural logarithm

I need to calculate such inequality. $ \ln\frac{2x}{x-1} \geq 0 $ I'm new to concept of $ \ln$ and clueless how to move on. Any tips?
0
votes
2answers
38 views

Proof that $H(X) \leq \log(|A|)$ (Shannon entropy)

The full question states: "Show that $$H(X) \leq \log(|A|)$$ with equality if and only if $P_X$ is uniform. Hint: use the Gibbs or log-sum inequality " I used "$A$" as the alphabet in here. My ...
1
vote
0answers
38 views

Integrate $\int_{-\infty}^\infty [4(\log r_1 - \log r_2) - 2(x_1^2/r_1^2 - x_2^2/r_2^2)]^2 dx$

As the title suggests, I am having trouble evaluating the following definite integral: $$\int_{-\infty}^\infty \left[4\left(\log r_1 - \log r_2\right) - 2\left(\frac{x_1^2}{r_1^2} - \frac{x_2^2}{r_2^...
2
votes
0answers
30 views

Prove $π(x+y)- π(x) \ll \frac y{\log (\log (y))}$ using Legendre sieve such that $10 ≤ y ≤ x$

How to prove $$π(x+y)- π(x) \ll \frac y{\log (\log (y))}$$ using Legendre sieve such that $10 ≤ y ≤ x$?
0
votes
1answer
24 views

Inequality in proof of 2nd Borel-Cantelli Lemma

At some point in the proof of the second Borel-Cantelli Lemma the the following inequality is mentioned: $$...=\exp\bigl ( \sum_{m=n}^k\log(1-P(A_m)\bigr ) \leq \exp \bigl (-\sum_{m=n}^kP(A_m) \bigr)$...