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Questions tagged [logarithms]

Questions related to real and complex logarithms.

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Exponents and Log

$3^{2x}-2\left(3^x\right)=3$ My solution: $\left(3^x\right)^2-2\cdot \:3^x=3$ Make the substitution $3^x=u$ $\left(u\right)^2-2u=3$ $u^2-2u-3=0$ $u=3,\:u=-1$ No solution for $3^x=-1$ $3^x=3$ ...
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13 views

How to get the depth of a recursion tree (help with logs)?

I'm reading this example from this site https://www.cs.cornell.edu/courses/cs3110/2012sp/lectures/lec20-master/lec20.html, and trying to understand the example: $T(n) = T(n/3) + T(2n/3) + n$. Mostly I'...
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1answer
43 views

Difference between $\log (x^2)$ and $2\log x$

Isn't $\log x^m$ same as $m\cdot \log x$? But the domain of let's say $\log x^2$ is negative infinity to positive infinity... on the other hand $2\log x$ is defined only for $x>0$ So given these ...
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2answers
24 views

How to simplify $\ln |\ln y | = \ln|x|+C$?

Let's say after solving an ODE, the solution I ended up with is as follows, $$\ln |\ln y | = \ln | x | + C$$ and I want to simplify it in a manner where I can get $y$ by itself. How would I go about ...
4
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3answers
99 views

convergence of the series $\displaystyle\sum_{n=1}^{\infty}\dfrac1n\log\left(1+\dfrac1n\right)$.

I am trying to check the convergence or divergence of the series $\displaystyle\sum_{n=1}^{\infty}\dfrac1n\log\left(1+\dfrac1n\right)$. My attempt: for a finite $p$,\begin{align}\displaystyle\...
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3answers
28 views

Proof by Induction of $\frac{d^n}{dx^n} (\ln(x))$ = $\frac{(-1)^{n-1}(n-1)!}{x^n}$ for $n\geq1$.

Conjecture for $\ln(x)$ and $\frac{d^ny}{dx^n}$ showed how to find the conjecture for $ln(x)$ after testing multiple values. For this question that I have, I have to prove using induction that $$\...
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1answer
28 views

Taylor expansion of $\log \frac{1+\exp(u+h)}{1+ \exp(u)}$

Hjort and Pollard write in their article Asymptotics for minimisers of convex processes that the following expansion holds for all $u$ and $u+h$, in terms of $\pi(u) = \exp(u)/\{1+\exp(u)\}$: $$\...
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1answer
40 views

Calculate $\sum_{n\ge2}\log\left(1-\frac1{n^2}\right)$

This is the expression whose sum I have to calculate: $$\sum_{n\ge2}\log\left(1-\frac1{n^2}\right)$$ I have tried to use the mengoli's series properties but I failed. The listed answer should be $-...
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1answer
48 views

$w = \log z$ is analytic on $D = \{z : \alpha < \arg z < \alpha +2\pi k\}$

$w = \log(z)$ is analytic on $D = {z : \alpha < \arg z = \theta< \alpha +2\pi k}$. Now I have been told, conceptually and without proof, that this is true. I have been shown that taking the ...
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0answers
21 views

Class of functions being continously logarithmically differentiable?

In analysis, one often talks about functions which are $C^1$,$C^2$,..., $C^\infty$ et.c. Does there exist some similar concept for how many times something is logarithmic continuously differentiable? ...
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91 views

Prove a strong inequality $\sum_{k=1}^n\frac{k}{a_1+a_2+\cdots+a_k}\le\left(2-\frac{7\ln 2}{8\ln n}\right)\sum_{k=1}^n\frac 1{a_k}$

For $a_i>0$ ($i=1,2,\dots,n$), $n\ge 3$, prove that $$\sum_{k=1}^n\frac{k}{a_1+a_2+\cdots+a_k}\le\left(2\color{red}{-\frac{7\ln 2}{8\ln n}}\right)\sum_{k=1}^n\frac 1{a_k}.$$ The case without $\...
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0answers
38 views

What is the cost of manufacturing per item?

A firm estimates that the total revenue, $R$, in dollars, received from the sale of $q$ items (in thousands of items) is $R = 100000 \log(1+1000q^{2}).$ The manufacturing cost $C$, in dollars, of ...
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1answer
33 views
+50

How to relate two series(GP and AP) using a positive real number, who have nothing in common?

Here is a question from my book: Given a GP and an AP with positive terms $a,a_1,a_2,a_3...a_n$ and $b,b_1,b_2,b_3,...b_n$ respectively. The common ratio of the GP is different from $1$. Then ...
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2answers
18 views

Interpret Logarithmic Values like Mean and Std

let's say i have a series of data containing prices on the log scale. How would i interpret a arithmetic mean of 0.55 and a std of 0.69 (both metrics are computed with the log prices). Is there an ...
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0answers
20 views

Solving an equation with different bases, each to an unknown power, but with a third value

This is a relatively straightforward question, but neither myself nor my colleagues are able to come up with a tidy solution without brute force: Represent $x$ algebraically, given the equation $$4^{...
2
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1answer
77 views

Find $\lim_{x \to 0^{+}} \frac{\pi^{x\ln x} - 1}{x}$ if it exists .

Let $f(x) = \frac{\pi^{x\ln x} - 1}{x}$ . Find $\lim_{x \to 0^{+}}f(x)$ if it exists . My try : $f(x) = \frac{\pi^{x\ln x} - 1}{x} = \frac{e^{x\ln x \ln \pi} - 1}{x}$ . Using $(\forall u\in\mathbb{R}...
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1answer
53 views

Logarithmic equation $\log_2(x+4)=\log_{4x+16}8$

So the problem goes: What is the product of all solutions in the equation $$\log_2(x+4)=\log_{4x+16}8$$ The solution to this should be $31\over4$, but I got $-14$. This is what I did: \begin{...
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0answers
16 views

Arithmetic Mean out of log values?!

i have an array of log-values (base e) and i'm struggling with interpretations of mean and standard deviation. For the true values i would calculate the arithmetic mean and std, if i do the same with ...
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1answer
17 views

Define function based on asymptotes and intercept

I'm looking for a function with the following characteristics: Vertical asymptote at $0$ (i.e. function never touches negative $x$-values) Horizontal asymptote at $7$ (i.e. function never results in $...
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2answers
38 views

The difference $\log(2)-\sum_{n=1}^{100}\frac{1}{2^n n}$ is

The difference $$\log(2)-\sum_{n=1}^{100}\frac{1}{2^n n}$$ is (1) less than $0$ (2) greater than $1$ (3) less than $\frac{1}{2^{100}101}$ (4) greater than $\frac{1}{2^{100}101}$ My Attempt: I ...
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4answers
80 views

Evaluating $\frac{\ln^2(6)}{\ln(5)\ln(7)}$ [closed]

Evaluate whether the ratio $\frac{\ln^2(6)}{\ln(5)\ln(7)}$ is greater than $1$. Has been giving me headaches; I'd highly prefer using pre-calculus notions, though any help is appreciated.
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33 views

Comparing Values -MAT-

I'd like to find out how a mathematician would think to solve this problem. This question has to be done without the use of calculators and using only basic assumptions. B. Which is the smallest ...
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0answers
38 views

Comparing $\log_5 6$ and $\log_6 7$ [duplicate]

I've tried comparing $\log_5 6%$ and $\log_6 7%$ with various methods(evaluating their ratio mostly) and haven't come to a conclusive proof to which one is greater.A hint I've received suggested ...
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4answers
73 views

Why is the domain of $\ln\left(\frac{x+1}{x-1}\right)$ different than $\ln(x+1)-\ln(x-1)$?

Given the two functions $$f(x) = \ln\left(\frac{x+1}{x-1}\right)$$ and $$g(x) = \ln(x+1)-\ln(x-1)$$ I can justify independently why $\text{dom}(f) = (-\infty, -1) \cup (1,\infty)$, and $\text{dom}(g)= ...
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3answers
52 views

Proof of algebraic equation

I have been trying to prove that this expression is true, but I don't think I have an adequate grasp of the rules of logarithmic expressions. Here is the expression: $$a^{\log_b c} = c^{\log_b a}$$ ...
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3answers
42 views

Proof of $\lim_{h\to 0}\frac{t^h-1}{h}=\ln t$ please

I should know this, but I don't. Please help me understand why $$\lim_{h\to 0}\frac{t^h-1}{h}=\ln t$$
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1answer
22 views

Question about proof that $Ln$ is analytic.

I have a question about a step in the proof of the following theorem. Theorem: $Ln: \mathbb{C}- \{x \in \mathbb{R}: x \leq 0\} \to \mathbb{C}: z = re^{i \phi} \mapsto \ln r + i \phi$ $(\phi \in (-\pi,...
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3answers
31 views

How to find the Graph of a Logarithm from two Points

I have a graph of a logarithm function with only two points marked. I need to find the equation for the function from these two points. The two points are $(8,3)$ and $(1,0)$. I already know the ...
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2answers
61 views

Prove $(1+x)\ln(1+x) + (1-x)\ln(1-x) \leq 2x^2$ [duplicate]

For all $-1<x<1$, prove: $$(1+x)\ln(1+x) + (1-x)\ln(1-x) \leq 2x^2$$ I am sure we should use the Jensen's inequality or the Taylor expansion of $\ln(1+x)$ and $\ln(1-x)$; however, I was not ...
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2answers
35 views

How to compare logs

I have a quick question about simplifying these exponents and then comparing them: $8^{\log_2 n}, 2^{3log_2(log_2n)}$ and $2^{(log_2(n))^2} $ I know the third one evaluates to $n^{log_2(n)}$, but I'...
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1answer
27 views

Stuck solving a logarithmic calculation

I'm preparing for my further studies (last year of high school, preparing so I can try and join the academy that I want), and just solving problems. Got stuck on this one: What is the value of: $$...
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1answer
43 views

maximising function subject to the constraint without using lagrange multiplier and other calculus techqniques

question: maximise the following function $f$ $$f= x^p y^q z^r$$ subject to the constraint $$ax+by+cz=p+q+r$$ i know how to do it using lagrange method of multipliers . but i'm looking for an ...
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1answer
25 views

The bacteria is increasing 5% per hour. 400 bacteria are present. Determine the equation that gives the number N of bacteria present after t hours. [closed]

How can I do this with full solution? What is the answer to this question? How could I determine an equation. This is all about exponential and logarithmic functions.
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2answers
27 views

Simplifying/Finding the natural log of two terms without logarithm laws.

If there's a natural log of two terms, which I cannot simplify with the laws of logarithms, how should I simplify it? e.g. $\ln(e^{6x} + 17)$ The full equation could be something like this: $$\ln(...
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3answers
94 views

Which is greater, $\left(\frac{e}{2}\right)^\sqrt{3}$ or $(\sqrt{2})^{\pi/2}$? (no calculators)

From a math contest in 1985: Determine which of the following is greater: (no calculators) $$\left(\frac{e}{2}\right)^\sqrt{3} \, \hspace{3mm} \text{or} \hspace{3mm} \, (\sqrt{2})^{\pi/2}$$ Hints ...
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152 views

Show that $\int_0^1\frac{\operatorname{Li}_3(1-z)}{\sqrt{z(1-z)}}dz=-\frac{\pi^3}{3}\log 2+\frac{4\pi}3\log^3 2+2\pi\zeta(3)$

Recently I have encountered some integrals involving Polylogarithms like this one or closely related integrals such as this one. Hence I am quite fascinated by these kinds of definite integrals - ...
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1answer
22 views

Rearrange the following equation into the form of $y=mx+c$ so the gradient can be used to determine the value of RC: $V=V_0(1-e^{-\frac{t}{RC}})$

Rearrange the following equation into the form of $y=mx+c$ so the gradient can be used to determine the value of RC: $$V=V_0(1-e^{-\frac{t}{RC}})$$ I've used logs to get it to $$\frac{RC(\ln V_0)}{RC ...
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1answer
37 views

proof for $n(1-\frac{k+1}{n})^{n\ln(k+1)/(k+1)}<ne^{-\ln(k+1)}$

I came across this inequality in a graph theory book, couldn't figure how to prove it. $$n\left(1-\frac{k+1}{n}\right)^{n\ln(k+1)/(k+1)}<ne^{-\ln(k+1)}.$$ $n$ and $k$ are both positive integers. (...
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1answer
28 views

The formula $P = 1,527,000 (1.015)^t$ gives the population $t$ years after 2008. Find the population in 2009 and 2010? [closed]

The formula $P = 1,527,000 (1.015)^t$ gives the population $t$ years after 2008. Find the population in (a) 2009 (b) 2010 What is the answer and a proper solution to this problem? This is all ...
2
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1answer
72 views

History of logarithmic function

The exponential function is a very important function and it arises naturally. For instance, consider the limit $\displaystyle \lim_{n \to \infty} (1+\dfrac{1}{n})^n$. The limit is evaluated to be ...
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49 views

Rules for equation involving addition and subtraction of logarithms

Given these two examples from my math course: Example A: $$\log(50) + \log(x/2) = 2 \implies \log 25x = 2 \implies 25x = 10^2 \implies x = 4.$$ Example B: $$\log(72) - \log(2x/3) = 0 \!\implies\! \...
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2answers
93 views

Show that $\int_0^1 \frac{\ln(x+1)}{x}dx=-\frac12\int_0^1 \frac{\ln x}{1-x}dx$

While doing a little bit research on the "alternating Basel Problem" I have come across this related post which states the equality $$\int_0^1 \frac{\ln(x+1)}{x}dx=-\frac12\int_0^1 \frac{\ln x}{1-x}...
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2answers
106 views

Solving $\log_{6}(2x+3)=3$. Can I start by dividing by $\log_6$?

For example, $$\log_{6}(2x+3)=3$$ The way I would go about this is solving for $x$. So we begin by dividing each side by $\log_{6}$: $$(2x +3) = \frac{3}{\log_{6}}$$ Then subtract $3$: $$2x = \...
0
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3answers
70 views

Simplify $(3\log x) - (2\log x)$ [closed]

How to simplify $(3\log x) - (2\log x)$? Would this become $(\log x )^ {\frac{3}{2}}$ or would this be just $3\log x-2\log x =\log x$? If so how to get $\log x$? I was given this question: solve for $...
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2answers
29 views

What is the expansion of $\log(N+x) = \log(N) + [\dots\text{blank}\dots] $? ($N \in \mathbb{R}+$ and $0 \leq x \leq 1)$.

I'm working on a math problem which might be solvable if I can re-express $\log(N+x)$ as $\log(N) +$ 'something. The problem I am having with the Taylor series expansion about $x=0$ is that it ...
1
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2answers
54 views

Maxima of product of two log functions

Is there a way to find the maxima of the product of two log functions. I need to find the maxima of: $f(x) = log\frac{1000}{x+k}.log(x+1) \text{, where k is a constant < 500, and } x \in [0,1000]...
5
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3answers
90 views

Is there any mistake in my approach for solving $ \int_0^{\pi/2} \frac{ \cos x}{3 \cos x + \sin x} \, dx $ ??

I had to evaluate this integral . $$ \int_0^{\pi/2} \frac{\cos x}{3 \cos x + \sin x} \, dx $$ Here is how I proceeded Dividing $N^r$ And $D^r$ by $\cos^3 x$ $$ \int_0^{\pi/2} \frac{ \sec^2 x}{3 \...
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0answers
27 views

How do I write $\sum\limits_{p \leq q \text{ prime}}\log (p-x)$ using Chebyshev's function(s)?

Question: If $x\in \mathbb{N}$ and $p$ and $q$ are prime numbers how do I write $\sum\limits_{p \leq q \text{ prime}}\log (p-x)$ using Chebyshev's function(s) ? I have some partial results: The case $...
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1answer
36 views

A query while showing that the Gamma function $\Gamma$ is logarithmically convex for $x \gt 0.$

We are using the general definition of gamma function defined on $\Bbb C \setminus \{\text{non-positive integers}\}$ i.e. $\Gamma(Z)=\frac {e^{-\gamma z}}{z} \prod (1+ \frac zn)^{-1} e^{\frac zn}$. ...
1
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1answer
51 views

How can I write $\sum\limits_{p \leq q \text{ prime}}\log (p-1)$ using Chebyshev's function(s)?

Question: If $p$ and $q$ are prime numbers how do I write $\sum\limits_{p \leq q \text{ prime}}\log (p-1)$ using Chebyshev's function(s) ? I would like to think $\sum\limits_{p \leq q \text{ prime}}...