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Questions tagged [logarithms]

Questions related to real and complex logarithms.

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0answers
22 views

Prove that $ \left \lfloor {\log n} \right \rfloor = \left \lfloor {\log \left \lceil \frac {n-1}{2}\right \rceil} \right \rfloor + 1$

We have been doing algorithm analysis in university, and after analyzing binary search algorithm, the following equation resulted. What we have to do now is to prove that $ \left \lfloor {\log n} \...
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1answer
26 views

Logarithm: octave and decade

I am having some difficulty of finding a common divisor between an octave and a decade. The doubling of a value say from $1$ to $2$ corresponds to a difference of $20\, \log_{10}(2) = 6.020600$ dB to ...
5
votes
1answer
79 views

Solving $9^{1+\log x} - 3^{1+\log x} - 210 = 0$ where base of log is $3$ for $x$

Question is to solve the equation for value of $x$. $$9^{1+\log x} - 3^{1+\log x} - 210 = 0; \quad \text{where base of log is }3$$ The answer given is $x=5$ I've tried to solve it. And got two ...
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1answer
25 views

Solve the equation for x: [on hold]

The question is Solve for x: log(x^2 -12x +36) = 2 The answer given is x = 16 or -4 Please solve it according to class 11
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2answers
22 views

Finding $f(z)=\log(\frac{(z+1)^2}{(z^2+4)})$ Laurent series

Determine the Laurent series of the function $f(z)=\log(\frac{(z+1)^2}{(z^2+4)})$ on the set $A=\{z|\:2<|z|\}$. I thought of using the following expansion: $\log(1+z)=\sum_\limits{n=1}^{\infty}\...
1
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1answer
27 views

Finding the logarithm to the nearest integer without a calculator

If I want to find the nearest integer of $log_2(1,000,000,000)$ What I tried was to use the change of base rule for logarithms, using base 10 should simplify this down. $$\dfrac{log_{10}(1,000,000,...
2
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1answer
52 views

Minimising quadratic objective subject to logarithmic inequality constraints

I need to minimise the following function: $$ (2a_1 + 2a_2 - 1)^2 + (2a_1 + 2a_3 - 1)^2 $$ subject to: $$ \sum a_i \log_2 a_i \geq -1 $$ where all the $i \in \{1,2,3,4\}$ and $a_i \in [0,1]$ and $\sum ...
-3
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2answers
36 views

All results of the equation x∈R [on hold]

This is the equation: $\ln(x+1)+\ln(x)=\frac{1}{2}\ln(49)$. I can't solve it. Could somebody help me and show me the way? Thanks a lot.
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2answers
27 views

Take Derivative Of Logarithm with Coefficient

IN my books, I see this example that derivative of $ln(\frac{1}{x^3})$ is $\frac{-3}{x}$. I know derivative of just $ln x$ is $\frac{1}{x}$. What is rule for multiplying coefficient with 1/x? I not ...
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1answer
52 views

Not Sure Why Limit Is In Book

I am looking at logarithms and derivatives. In my books, Bostock and Chandler, it saying: $\frac{dy}{dx} = \lim_{\delta x \to 0} \frac{\delta y}{\delta x} = \lim_{\delta x \to 0}(\frac{1}{\frac{\...
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2answers
32 views

Question relating to logarithms

Hello I have the following question I have to find the answer of this logarithmic formula $$(\log(2^5) + \log(4^{0.2}))\times(\log(5^2) + \log(25^{0.5})) .$$ I am currently having problems with the ...
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0answers
28 views

Series - convergence - logarithm [on hold]

I need help for: $$\sum_{n=2}^{\infty}{\frac{ 1}{(n\ln(n)\sqrt{\ln(\ln(n))}}}$$ Does it converges?
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0answers
44 views

How to solve mixed power and exponential equation?

The exercise says that I should find "real zeros" of the following function: $$f(x)=7-2x^2+2^{-x}$$ I have tried several ways, but I keep getting stuck as I take log-s of both sides. Can I solve ...
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1answer
28 views

Domain of $2\log_2x$ and $\log_2x^2$

Domain of $2\log_2x$ and $\log_2x^2$ For $2\log_2x$ domain is simply $x>0$ For$\log_2x^2$ domain may be $R$ , I think so. Is am I right or wrong .
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2answers
27 views

Log variable base with variable

Looking for help with this equation. Trying to help boyfriends younger sister but answer is either all numbers or its impossible: $$ \log_x \left(x^5\right) = 5 $$
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0answers
23 views

When $N^b (\log N)^c = 2^N$

What b and c should equal to, to say that this equation is correct(the base of log is 2): $$ N^b (\log N)^c = 2^N $$ *I'm a beginner in Math and need this to be done to go further. Would be nice ...
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1answer
47 views

How is $\log(x) \leq x-1$ for all $x>0$?

sorry if this is a very obvious question, but I'm writing a proof for the Kulback-Leibler inequality and the first step is to state $\log(x) \leq x-1$ for all $x>0$. I get it for $x>1$, but in ...
2
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3answers
51 views

Find the sum: $\sum_{n=1}^{\infty} \ln\left(\frac{b+n+1}{a+n+1}\right)$

Let $0<a<b$, I would like to compute the sum $$\sum_{n=1}^{\infty} \ln\left(\frac{b+n+1}{a+n+1}\right).$$ But first I am worrying that a test convergence might lead to the divergence of this ...
1
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2answers
42 views

Logarithmic Differentiation of $x(3x^2-9)$

I am trying to teach myself how to use logarithmic differentiation, but I'm not getting the right answer for some reason. I think it is because I may be improperly using log rules, but I can't find ...
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4answers
54 views

Find the value of $\sum_{r=1}^4 \log_2 (\sin(\frac{r\pi}{5}))$

Find the value of $$\sum_{r=1}^4 \log_2 (\sin(\frac{r\pi}{5}))$$ My apporach:- $$\sum_{r=1}^4 \log_2 (\sin(\frac{r\pi}{5}))$$ $$=\log_2 (\sin(36^{\circ}))+\log_2 (\sin(2*36^{\circ}))+\log_2 (\sin(3*...
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2answers
19 views

Is there a way to visualize how taking the logarithm of a function doesn't change the locations of the max and min?

I was attempting to take the second derivative of the binomial distribution with respect to p to show that p = j/n is a maximum (where the binomial coefficients are $\binom{n}{j}$). From this answer ...
2
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2answers
26 views

Log Functions Inside Absolute Value

Is the function below always positive for $0< x <1$? (I am determining if the function requires the modulus sign or not.) $$\frac{1}{2}\log\left|\frac{1+\log(x)}{1-\log(x)}\right|$$ My first ...
2
votes
2answers
61 views

Prove that if $e<y<x$ then $x^{y}<y^{x}$

Prove that if $e<y<x$ then $x^{y}<y^{x}$ My try:I tried to use Taylor's theorem: $$y^{x}-x^{y}=e^{xlny}-e^{ylnx}=1+xlny+o(xlny)-1-ylnx-o(ylnx)=$$ $$=lny^{x}-lnx^{y}+o(xlny)-o(ylnx)=ln\frac{y^...
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1answer
28 views

Solving simultaneous logarithmic equations from Newton's law of Cooling

A cup of warm water at $46$ degrees is placed into a refrigerator. 10 minutes later, the water is $39$ degrees, and another 10 minutes later, the water is $33$ degrees. Use Newton's law of cooling ...
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2answers
47 views

$\frac{\left(10^4\right)}{x^2}=\frac{\left(x^{\left(8-2\log x\right)}\right)}{10^4}$ Solve for x.

$\frac{\left(10^4\right)}{x^2}=\frac{\left(x^{\left(8-2\log x\right)}\right)}{10^4}$ Solve for x. $\frac{\left(10^4\right)}{x^2}=\frac{\left(x^{\left(8-2\log x\right)}\right)}{10^4}\Rightarrow 10^8=\...
2
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2answers
89 views

Is there a way to calculate the improper integral $\int_0^\infty \big(\frac{\ln x}{x - 1}\big)^2 dx$?

Is there a way to calculate the improper integral $\int_0^\infty \big(\frac{\ln x}{x - 1}\big)^2 dx$? What have I tried: $$\int_0^\infty \Big(\frac{\ln x}{x - 1}\Big)^2 dx = \int_0^1 \Big(\frac{\ln ...
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1answer
24 views

Fitting a logarithmic trendline on already logged values

This is the situation. I am running trials with a population simulator, which produces various outputs, with the variance of these outputs being dependent on the number of clones (recursions) the ...
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2answers
26 views

exponents and logarithms question

Find the sum of all solutions to \begin{align*} (\log_2 x)(\log_3 x)(\log_4 x)(\log_5 x) &= (\log_2 x)(\log_3 x)(\log_4 x) + (\log_2 x)(\log_3 x)(\log_5 x) \\ &\quad + (\log_2 x)(\log_4 x)(\...
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2answers
39 views

How do I take the $\log$ of a inequality of the form: $0<x<1$?

$0<x<1$ taking the $\ln$ would give $\ln0<\ln x< \ln 1 \implies \text{undefined} < \ln x <0 $. If $x=1/2$ I could use $1/4<1/2<1$ and would be ok. But I only know that $x>...
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1answer
29 views

change to logarithmic form. solve for P.

change to logarithmic form. solve for P. (2/3)log(R) + 0.05 = log(P) the base is 10 for both logs. How do I answer this?
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2answers
29 views

The numerical value of this determinant is?

For positive numbers $x$, $y$ and $z$, the numerical value of the determinant \begin{vmatrix} 1 & \log_xy & \log_xz \\ \log_yx & 1 & \log_yz \\ \log_zx & \log_zy & 1 \\ \...
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0answers
19 views

Interpret model estimates after log transformation

I sat up a mixed-effects linear model with the dependent variable log-transformed (in oder to get it normal distributed and as ist is common with this kind of data in other publications). All fixed ...
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5answers
63 views

What is $\log_{a}{x} \cdot \log_{y}{a}$ given below system of equations?

I let $\log_{a}{x}=m$ and $\log_{y}{a}=n$. So I have to find $m\cdot n$. From the system of equations we get $$m-\frac{1}{n}=1 \quad \quad n-\frac{1}{m}=1$$ From here I find that $m=n$ (...
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2answers
33 views

How to mix and match log rules?

I know that $\ln(xy) = \ln(x) + \ln(y)$, that $\ln(\frac{x}{y}) = \ln(x) - \ln(y)$ and that $\ln(\sqrt{x}) = \frac{1}{2}\ln(x) = \frac{\ln(x)}{2}$ but I can't seem to find any rules for more complex ...
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3answers
54 views

How to solve exponential equation of $x + e^{-x }= 3$?

I've got the following equation: $\lambda + e^{-\lambda} = 3$ $3 - \lambda = \frac{1}{e^{\lambda}}$ $\frac{1}{3-\lambda} = e^{\lambda}$ Now, I take the natural log of both sides: $ln(\frac{1}{...
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1answer
32 views

log-transforming and finding partial derivatives

I trying to set up an economic model of trade, and i have a question regarding the partial effect of one of my variables that is logged along with my general function $X$. My model of bilateral ...
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1answer
79 views

Prove that the limit of the sum is $\ln(2)$ [closed]

I need to prove that $$\lim_{n\rightarrow\infty}\sum_{i=1}^{n}\frac{\left(-1\right)^{i+1}}{i}=\ln2$$ There is a hint I can use that says $$ \sum_{i=1}^{2n}\frac{\left(-1\right)^{i+1}}{i}=\sum_{i=...
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1answer
40 views

Range of a log function

Find the range of $f(x)=\ln(\frac{\sqrt{8-x^2}}{x-2})$ My attempt :- $$f(x)=\ln(\frac{\sqrt{8-x^2}}{x-2})$$ $$=0.5\ln(8-x^2)-\ln(x-2)$$ After this I can't solve and not able to find the range.
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3answers
50 views

Proof and Geometric intuition of $u \leq 2\ln(1+u)$ for $u \in [0,1]$.

How can I prove the inequality $$u \leq 2\ln(1+u)$$ for $u \in [0,1]$. What is the geometric intuition behind this inequality?
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2answers
51 views

Prove that $\log(x^{\frac{n}{m}}) = \frac{n}{m}\log(x)$

I need to prove the following: $$\log(x^{\frac{n}{m}}) = \frac{n}{m}\log(x)$$ In order to prove it I can only use the definition: $$\log(x) = \int_1^x{\frac{1}{t}dt}$$ I tried using some change of ...
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1answer
50 views

Find and classify the critical points of $f(x) = \ln(e^{3x} −2e^{2x} + e^x + 2)$, then sketch its graph

Question : Find and classify the critical points of $$f(x) = \ln(e^{3x} −2e^{2x} + e^x + 2),$$ then sketch its graph I have found out that: first derivative = $( 3e^{3x} - 4e^{2x} + e^x )(e^{3x} −2e^...
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1answer
18 views

How do I directly show that Log is discontinuous on the interval $\{-\infty<\Re z\le 0\}?$

Given the knowledge that the exponential function maps bijectively each half-open horizontal strip of width $2\pi$ onto $\mathbb C\setminus \{0\}.$ So, in particular, if we restrict it to $-\pi<\...
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1answer
32 views

How to calculate basic log expansion

Without a calculator, so I cannot use the simple expansion of log(1+x) Using wolfram we can see that $\log(\frac{\sqrt{n}}{2t}[\exp(\frac{2t}{\sqrt{n}})-1)])$ $=\frac{t}{\sqrt{n}}+\frac{t^{2}}{6n}+....
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1answer
40 views

Efficient and accurate approximatiion of logarithm of binomial coefficients

I am searching for an efficient and accurate way to approximate the logarithm of binomial coefficients since I have to deal with extremely large numbers in C++. Using Stirling approximation, I am able ...
2
votes
2answers
73 views

Why do we take $(1+z)^{\alpha}$ as $e^{\alpha \operatorname{Log}(1+z)}$?

Let $\alpha$ be a complex number. Show that if $(1+z)^{\alpha}$ is taken as $e^{\alpha \operatorname{Log}(1+z)}$, then for $|z|< 1$ \begin{equation*} (1+z)^{\alpha} = 1 + \frac{\alpha}{1}z + \...
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vote
2answers
54 views

$0<e^{-\sqrt x}<\frac{1}{x^a}$ inequality

How can we prove that $0<e^{-\sqrt x}<\frac{1}{x^a}$ is true for all $x>B(a)$ for some $B(a)$ and some fixed $a>0$. By the mean theorem I can only receive $\frac{1}{1+\sqrt x}>e^{-\...
0
votes
1answer
32 views

Logarithmic pattern from 0 to 1 to calculate probability. [closed]

I'm sorry if this is badly explained, I'm really a computer programmer. I have 13 different variables all initially assigned to a integer of 0.5. thr = 0.5 ...
0
votes
0answers
91 views

When is $f^{p-1}(1)\equiv 1 \bmod f^{\log_mp}(1)$?

If m is coprime to p ( p prime) and $f(x)=x\cdot m$ this is the functional equivalent of Fermat's little theorem. When else is this true ? I know 1 is the multiplicative identity. Okay, so f(x)=1 is ...
0
votes
2answers
24 views

Big O Hierarchy log log 16 less than 1

My lecture has given me a Big O hierarchy table that shows $1 \leq \log(\log(16))$. How is this possible given $\log(\log(16)) = 0.08066976367$? More specifically, $1 \leq log(log(n))$ for all $n\...
0
votes
1answer
21 views

Dealing with log functions

I have a function where all variables are linear except for the log function. In one equation I have $\log(x)$ and another equation I have $\log(1-x)$. How can I linearize $\log(x)$ and $\log(1-x)$?