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Questions tagged [logarithms]

Questions related to real and complex logarithms.

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2answers
16 views

An inequality flip involving logarithms base 0.92

The question here involved you needing to find the smallest integer $ n $ such that $S_n > 72$ where $$S_N = 6\sum_{i=1}^n 0.92^{i-1} $$ This seemed like a basic question with nothing unusual ...
-1
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3answers
39 views

How to prove $\log_{18}12$ is irrational?

I was trying to prove this through contradiction where I supposed that it was rational but it didnt seem to work out. Any help would be appreciated.
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1answer
13 views

Using a single function, how can I model a logarithmic-like increase followed by an exponential-like decay?

Using a single function, how can I model a logarithmic-like increase followed by an exponential-like decay? The transition will occur at some critical value, Tcrit. A graph of my experimental data is ...
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3answers
48 views

Expressing $\log_4 0.75$ in terms of $\log_43$ and $\log_45$ [on hold]

Given that $$\log_43 = a$$ $$\log_45 = b$$ How to find $\log_4 0.75$ in terms of $a$ and $b$?
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1answer
25 views

Solving for k in $ \log_{5} x = k \cdot \log_{10} x$

If $ \log_{5} x = k \cdot \log_{10} x$ find the value of $k$ (rough approximation) without using calculator. what i did is: $ \log_{5} x = \log_{5}10 \cdot \log_{10}x$ $\therefore \:k = \log_{5}...
1
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1answer
36 views

How to prove that $\log \log n/ \log n$ goes to zero with almost elementary techniques?

I am trying to show that: $$ \lim_n \frac{\log \log n}{\log n} = 0 $$ without using "advanced" calculus techniques (derivatives, analysis of the function $x \mapsto \log \log x/ \log x$...). I was ...
1
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0answers
41 views

how many real solutions does the equation $~2^x+8^x=2\cdot5^x~$ have? LOG EQUATIONS [on hold]

How many real solutions does the equation $~2^x + 8^x = 2 \cdot 5^x~$ have? The answer is $~3~,$ and I need an explanation.
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3answers
34 views

Log of e raised to an exponent

I have a textbook which states the following: $y=e^{(-\lambda x)}$ it then takes the log of both sides and comes up with: $log \ y = - \lambda x$ Why is the right side what it is? Shouldn't it be: ...
-2
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2answers
26 views

Logs question unknown powers [on hold]

Can somebody help me with solving for x here. 3^x = 5^(x-2)
1
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2answers
42 views

Need help in how to approach exponential equations

How do I approach solving these types of equations $10^x -5^{x-1}×2^{x-2}=950$
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1answer
45 views

What is the value of e to the power log (1+x)? [on hold]

What is the value of e to the power log (1+x) given that, we and perhaps I also know the value of $e^{\log(x)} = x$ and $e^{\log(x^{-1})} = 1/x$ since $1/e^{\log (x)}$ is same as $1/x$?
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1answer
18 views

Is $2^x+x$ Exponential Function and $\log_2(x)+x$ a Logarithmic Function?

I just wonder can we call $2^x+x$ an exponential function or not? With the same way of thinking is $\log_2(x)+x$ a logarithmic? or we can't have another term with theses functions holding a ...
0
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3answers
24 views

I have to determine (without solving the problem) an interval in which the solution of the given initial value problem is certain to exist.

The equation is the following: $$(t-6)y' + (\ln t)y = 4t,~~~~~~ y(1) = 4$$ I really tried do understand how to do it but I failed and there is nothing about it in my notebook so any help is ...
-2
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1answer
32 views

Prove that log (a^n)=nloga, n is an irrarional number [on hold]

Proof of the basic log rule, easy to work out when n is an integer or fraction but what to do when n is irrational?
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2answers
44 views

prove logarithmic function is big oh of non logarithmic function

I am having trouble knowing how to find out whether a given logarithmic function, |f(x)|, is an element of the set of all functions less than or equal to C * |g(x)| where g(x) is a non-logarithmic ...
3
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1answer
36 views

Asymptotic expansion of $\ln (1-\ln(\varepsilon))$

Question: Find an asymptotic expansion for $\ln (1-\ln(\varepsilon))$ as $\varepsilon \rightarrow 0$. Attempt: I don't even understand what exactly I am supposed to do. I am not given an ...
2
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0answers
85 views

Beautiful refinement of Am-Gm.

I see on a french forum the following inequality : Let $a_i>0$ be $n$ real numbers such that $\sum_{i=1}^{n}a_i\ln(a_i)=0$ then we have : $$n\Big(\prod_{i=1}^{n}a_i\Big)^{\frac{1}{n}}\leq n\...
3
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3answers
41 views

Proper syntax for summation of logarithms

In my textbook I found: $$\sum_{k=1}^{n}\ln{(k)}=\log(1)+\log(2)+\log(3)+\cdots+\log(n)$$ Shouldn't it be: $$=\ln(1)+\ln(2)+\ln(3)+\dotsb+\ln(n)\;\text?$$
3
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3answers
41 views

Tricky log equation involving products of logs [on hold]

Find the x values that satisfy the equation $$\log_2x\log_3x\log_5x=\log_2x\log_3x+\log_2x\log_5x+\log_3x\log_5x$$ I am unable to start off this question. Any suggestions on how to start it off ...
7
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1answer
83 views

Prove inequality $|y \ln{y} - x \ln{x}| < 2 |\ln{\frac{1}{|y-x|}}|$ when $x,y \in (0,1]$, $x \neq y$.

EDIT: Counter-example found. Statement is FALSE. However, I think the argument still has value. It is true if you restrict the domain to $[0.223,0.716]$. Maybe $[\frac{3}{10},\frac{7}{10}]$ so it’s ...
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0answers
14 views

Maximize a limit, logarithmic utility

We have a game with $m$ different outcomes, labeled $1,2...m$. Which each occurr with probability $p_1,p_2,...p_m$. If outcome $i$ occurrs you get m-times what you bet on that outcome back. Every ...
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0answers
14 views

Does a “log-growth” contribution have any meaning?

Imagine that a company's sales increase in a given year, due to an average selling price increase and a volume increase (the problem would be equivalent to that of GDP growth in terms of Per capita ...
1
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2answers
38 views

Is $\lim_{b\to\infty} (x\log_b(x)) = 0$?

Is $$\lim_{b\to\infty} (x\log_b(x)) = 0$$? I started to attempt this problem by graphing $y=x\log_2(x)$ and kept increasing the base of log to see what's the behaviour. You can see here that as the ...
2
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1answer
37 views

Eigenpairs of $A$ and Eigenpairs of $\exp(A)$

I would like to understand under which assumptions the following statement holds \begin{equation} (\lambda,v) \text{ is an eigenpair of the matrix } A \Leftrightarrow (\exp(\lambda\Delta),v) \text{ is ...
-1
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1answer
21 views

subject, logarithms, I developed a part. [closed]

Solve the following : $$(a.) \space \space \space 8 ^{\log _2 ^5}$$ $$(b.) \space \space \space 3 ^{1 + \log _3 ^4}$$ I developed a part, but I couldn't put the image too ...
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2answers
33 views

Solving logarithm equation of $\log_{3}(5x-2)+\log_{3}(x)=4$

My work so far is $$\log_{3}(5x-2)+\log_{3}(x)=4$$ $$\log_{3}(5x-2)+\log_{3}(x)=\log_{3}(81)$$ $$\log_{3}\left(x(5x-2)\right)=\log_{3}(81)$$ $$5x^2-2x-81=0$$ Is it correct so far ? Thanks for your ...
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1answer
13 views

Re-arranging with natural logs to different powers

I am trying to rearrange the equation $$T=\frac{1}{A+B\cdot \ln(R)+C\cdot \ln(R)^3}$$ Where $A, B\space and\space C$ are constants and $R$ is the independent variable. I would like to get an equation ...
0
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1answer
50 views

How to determine without calculator which number is bigger?

I get 4 scores and I need to sort them from most relevant(biggest score) to the least. 1)$$\frac{3}{9}\log(\frac{1}{4})$$ 2)$$\frac{3}{12}\log(\frac{1}{4})$$ 3)$$\frac{3}{8}\log(\frac{1}{4})+\frac{...
3
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3answers
79 views

Showing that $\ln \frac{1+e^x}{1+e^{-x}} = x$

Is the below a valid approach? $$\frac{1+e^x}{1+e^{-x}} = \frac{1+e^x}{1+e^{-x}} \times \frac{e^{-x}}{e^{-x}}$$ We know that $\frac{e^{-x}}{1+e^{-x}} = \frac{1}{e^x+1}$, so $$\frac{1+e^x}{1+e^{-x}} ...
-1
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1answer
15 views

Level curve with function containing natural logarithm

How do I sketch a level curve that has natural logarithm in its function? For example: $Z(x, y) = ln(xy) − x$ when $x > 0$ and $y > 0$ I can't find anything about it, so if you have a source ...
1
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4answers
66 views

Root of exponential equation

I am trying to find the roots of the equation $$ e^{x} -\cos x = 0. $$ Used the Lambert W function to arrive at $$ x = W(x\cos x), $$ but I don't know how to proceed from there to get the explicit ...
1
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1answer
49 views

Solve for $x$: $\frac{1}{\log(x+2)^2}+\frac{1}{\log(x-2)^2} = \frac{5}{12}.$ [duplicate]

Solve for $x$: $$\frac{1}{\log\big((x+2)^2\big)}+\frac{1}{\log\big((x-2)^2\big)} = \frac{5}{12}.$$ My Attempt: \begin{align*} & \frac{1}{\log(x+2)^2}+\frac{1}{\log(x-2)^2} = \frac{5}{12} \\ \...
0
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3answers
27 views

Prove Identity involving logarithms and exponents

Can you give me some hints on how to solve the following identity? $a^{\ln(n)} = n^{\ln(a)}$
0
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1answer
33 views

What's the purpose of Log. And natural log.? [closed]

I am trying to learn about logarithm and nat. Logs. , I was curious however since where I am reading there isn't really an explanation but, why and how do we use logs?
4
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4answers
57 views

Series for $\log 3$

I have the following series: $$\sum_{k=0}^{\infty} \left(\frac{1}{3k+1}+\frac{1}{3k+2}-\frac{2}{3k+3}\right)$$ Wolfram says this is just $\log 3$. I have been trying to figure out how this works ...
2
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2answers
24 views

The equation $100\log(5x)\log(2x)+1 = 0$ has two distinct real roots $\alpha$ and $\beta$. Find the value of $\alpha\beta$.

The equation $100\log(5x)\log(2x)+1 = 0$ has two distinct real roots $\alpha$ and $\beta$. Find the value of $\alpha\beta$. I'm having trouble with this because the answer key says $1/10$ as the ...
2
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0answers
14 views

Why is the expected gradient of a density not parallel to the expected gradient of the log density?

I'm cross posting from Stats Stackexchange, in case this community has more insight - hopefully that's okay! I'm confused by a seemingly counter-intuitive property of the interaction between ...
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3answers
34 views

sequence & series # logarithmic series question

The series 2[$\frac{1}{3x+1}$ + $\frac{1}{3(3x+1)^3}$ + $\frac{1}{5(3x+1)^5}$ + ...] is equal to
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4answers
70 views

Compute $\lim\limits_{x \to \infty} \frac{\ln (x!)}{x \ln (x)}$

I was just playing with logarithms and factorials, and then realized that $f(x) = \frac{\ln (x!)}{\ln(x)}$ is slower than $f(x) = x$. From there I got $\frac{\ln x!}{(x \ln (x))}$. In most calculators,...
0
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1answer
41 views

How is $\int_0^1 \ln(\frac{1}{1-x})dx=1$ using series expansion?

How is $$\int_0^1 \ln \left(\frac{1}{1-x} \right) dx=1$$ using series expansion? This is simple if one integrates it directly by first noting that the integrand is same as $-\ln(1-x)$, which can ...
1
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0answers
36 views

A book states that $\ln(2e^2)$ is the same as $\ln(2\cdot 2^2)=\ln8$. Why?

In a book it states $\ln(2e^2)$ is the same as $\ln(2\cdot 2^2)=\ln8$. When I reduce the expression $\ln(2e^2)$, I get the result $\ln2+2$. This seems right, since you use logarithm rule $\log(ab)=\...
1
vote
1answer
25 views

Closed formula or approximations for $\sum_{i=2}^{n} \lfloor \log_{i}(n) \rfloor$

I was working on this problem: How many integer pairs $ 2\le x, y \le 2019$ exist such that $\log_x(y)$ is an integer? I found this to be $$\sum_{i=2}^{2019} \lfloor \log_i(2019) \rfloor = 2086$$ ...
0
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0answers
24 views

Finding the model of a logarithmic dataset

I'm trying to find the model that best fits the oxidation diffusion in particular type of metal which depends on a constant temperature and linear time. Here are both tables: https://pastebin.com/...
-1
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5answers
68 views

Why can we calculate log (base 10) of any natural number? [duplicate]

I am curious to know the concept behind. Let's say y = 10^x So, y should be a number with zeroes but y is 10 multiplied to 10 x times. But this is not the case. For eg 10^3.5 = 3162.27766017 So, ...
0
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1answer
53 views

Can't find the set of solutions

Set of solutions of this inequality : $$\log_2({ \space x ^2 - 1}) < 1$$ The answer given is :$$ \sqrt3 < x < - 1 \cup 1 < x < \sqrt3 . $$
4
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1answer
49 views

How to solve this equation for x?

can somebody help me with this? $$2=2^{1-x}+(\frac{2}{3})^{1-x}$$ I guess I somehow have to isolate this 1-x term and then use the ln. But I don't get how.. Thanks in advance! Mostly irrelevant ...
0
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0answers
22 views

Using "ceil' To Make A function With seamless Changes

I'm trying to use ceil and or floor to accomplish something arrived at in another post. See lower two images for details: The uppermost image (of the two) is my question and the next is the answer. ...
2
votes
2answers
47 views

Integral related to the softplus function

Let $$ f(x) = \log(1+e^{2x+1}) - 2\log(1 + e^{2x}) + \log(1 + e^{2x-1}). $$ According to Wolfram Alpha, $$ \int_{-\infty}^\infty f(x)\,dx = \frac 12.\tag{$*$} $$ $f(x)$ is a "bump function" built out ...
-2
votes
1answer
17 views

Need some help for logarithms please!

Given $\log_2 P=x, \log_2 Q=y,$ and $\log_2 R=z$, determine $$\log_2 \frac{R^2\sqrt{Q}}{P^3}$$ interms of $x,y$ and $z$ Could someone please take me through the steps on how to solve this. Thank you!
4
votes
2answers
47 views

Logarithmic equation, some variables in bases and in arguments

This is the exercise, there are no clues in the book about it. $$ 40\log_{4x}x^\frac{1}{2}-14\log_{16x}x^3=-\log_{\frac{1}{2}x}x^2 $$ Solutions given by the book: $x=1; x=4; x=\frac{\sqrt{2}}{2}.$ ...