Questions related to real and complex logarithms.

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1answer
431 views

Why are there two series representations of the natural logarithm?

On the Wikipedia article of the natural logarithm one finds two different series representations for $\ln(x)$: $\ln(x)= (x - 1) - \frac{(x-1) ^ 2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4} \cdots$ ...
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2answers
332 views

Logarithm question for Algebra 2/Trig class

$$\frac{1}{2} \log(x+2)=2$$ I'm decently good at logarithms but this one seems to be tricky, when I did it myself I got a negative decimal as my answer but I'm not 100% confident in it, and I would ...
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0answers
11 views

PDF of the logarithm of a chi-squared random variable

Could someone give me a hint, what could be the expression of the PDF of the following random variable Y: Y = a*log(b+X), where a,b are constants and X is a noncentral chi-squared distributed random ...
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2answers
41 views

How to differentiate $\ln(a^x)$?

Can someone give me the process to differentiate this (with respect to $x$)? $$ \ln(a^x) $$
8
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5answers
327 views

Natural logarithms base $e$

Why is $e$ used as a base of natural logarithms everywhere? Is the origin from the fact that exponential is the only function with the unique property of its differential and integral same and that ...
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2answers
39 views

Solving for $x$ using $\ln$ or any possible way.

$$ 12.46x=1-(1+x)^{-20} $$ I tried solving for $x$ using $\ln$ and other methods but the only answer i got was 0.8. The correct answer is approximately to $0.05$.
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vote
1answer
57 views

Who are the two men credited with inventing logarithms?

This is a bonus question on a pre-calculus quiz I've been tasked with grading. Napier is clearly one of the answers. Who should I accept for the second inventor? In particular, should Newton be ...
5
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1answer
73 views

Lower estimate for $(\frac{\ln(1+2x)}{\ln(1+x)}-1)(1+2x)^{1/2}$ where $x>0$

I want to prove that: $$\left(\frac{\ln(1+2x)}{\ln(1+x)}-1\right)(1+2x)^\frac{1}{2}\geq 1$$ where $x>0$. Any help appreciated. Thanks!
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1answer
44 views

How I can solve this equation with respect to the variable $t$?

How I can solve this equation with respect to the variable $t$? $$\left\lfloor{\frac{\ln(t+1)}{\ln 2}}\right\rfloor=\left\lfloor{\frac{\ln t}{\ln2}}\right\rfloor+1$$ where $\left\lfloor {y} ...
3
votes
2answers
30 views

Solve the recurrence relation by taking the logarithm of both sides and making the substitution $b_n = \lg a_n$

Solve this recurrence relation: $$a_n = \left(\frac{a_{n-2}}{a_{n-1}}\right)^{\frac{1}{2}}$$ by taking the logarithm of both sides and making the substitution $$b_n = \lg a_n$$ A couple years ago ...
2
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1answer
34 views

How to get the peak value of this logarithmic equation?

Is there a way to get the peak point of the following equation? $$ (a_1-a_2 x)\ln\left(1+\frac{b_1 x}{b_2 x+b_3}\right),$$ where $a_1,a_2,b_1,b_2,b_3$ are all positive constant values and $x$ is also ...
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vote
1answer
15 views

Is this change of variable correct?

Take the following function: $$\frac{dn}{d[\log(x)]} = a\exp{\frac{-(\log(x) - b)^2}{c}}$$ I'm interested in obtaining the form for $dn/dx$, so I take: $$\frac{d[\log(x)]}{dx} = \frac{1}{x} ...
2
votes
1answer
33 views

Cauchy Principal Value for log integral

How do I evaluate the expression $\lim_{\xi\to0}(\int_0^\xi\! \ln(\frac{1}{r})\frac{F}{\xi} \, \mathrm{d}r) $ , where$\ F,\xi $ are real numbers and $\xi\geq0$. Integration gives the expression ...
0
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0answers
39 views

Summation of logarithms

I am trying to calculate the sum $\ln(a-x_1)+\ln(a-x_2)+....+\ln(a-x_n)$ and solve it somehow with respect to a ($x_1,x_2,....,x_n$ are measurements of a simulation) . The number of terms in the sum ...
0
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1answer
30 views

Logarithm help - change to exponential form, solve.

I need help with number 55 and 57 here. I need to change to exponential form and solve, but I can't seem to figure them out. Thanks! Solve for $P$ $$\dfrac23\log R+0.05=\log P$$
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1answer
44 views

Complicated Logarithm

If $x>0$, $y>0$, and $$x^2 + y^2=98xy$$ then $\log(x+y)$ can be expressed as $A\log(x)+B\log(y)+C$ where $A,B,C$ are real numbers and all logarithms are base $10$ logarithms. Compute $100ABC$. ...
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3answers
32 views

series involving a logarithm of ${1\over ln^2(n)}$

$${\sum_{n=2}^\infty}= {1\over ln^2(n)}$$ Can I substitute ${x}$=${1\over ln(n)}$ and using the integral test, set it up to be $${\lim_{t\to infty}}= \int_2^tx^2 dx$$ and solve from there? and then ...
0
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3answers
23 views

the difference between these two logarithm

I was just wondering what is the difference between ${1\over \ln(n^2)}$ and ${1\over \ln^2(n)}$ I know that ${1\over \ln(n^2)}$ is ${1\over 2\ln(n)}$ through the power rule, but I am not so sure ...
0
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3answers
29 views

Solve(Using Logarithms)

Another problem I can't quite figure out: ${1\over2}\ln(a+1) + \ln 5 = 1$ Solve using logarithms. Thank you!
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3answers
38 views

Solve the equation(using logarithms)

I'm having trouble with this math problem: Solve the equation(using logarithms) $7^{2x+1} = 5^x$ Thanks!
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3answers
36 views

Logarithm - Convert to exponential form

this needs to be converted to exponential form and I can't seem to figure it out. Any help is appreciated, thank you! $$10 \log(1+i) = \log 2$$
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0answers
79 views

Solving an exponential equation without the quadratic formula

High school math student here. In my homework I was asked to solve $16^x +4^{x+1} - 3= 0$ and I used substitution to get $x=\log_4{(-2+\sqrt7)}$. However, this was in the chapter on ...
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3answers
52 views

Limit question $\infty^{0}$ type

$$\lim_{x\to\frac{\pi}{2}^-} (\tan x)^{\cos x}$$ I just tried to write $e^{\ln(\tan x^{\cos x})}$ form but I couldn't solve the limit.
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1answer
29 views

Understanding the transformation of values when plotting logscale in Matlab.

I'm playing with some transistor test data and having trouble understanding what is probably a very basic principle. Below are my two graphs, plotted linearly and then with a logarithmic scale on the ...
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0answers
28 views

Determinant from matrix of logarithms

Is there a way to get the determinant $\text{Det}(M)$ of a matrix $M$ from the matrix of its logarithms, i.e. $\Bigg( \begin{smallmatrix} \log(M_{00}) & \log(M_{01}) & \ldots \\ \log(M_{10}) ...
2
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1answer
64 views

How to find the general sum formula of this logarithmic series $\log 5+\log 5+ \log 605+\log 6655+\dots$

I have another question about series. Now, this is about series involving logarithm. In the previous post, we can easily grouping the same factor from this series: http://tinyurl.com/kbg26ye But, ...
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3answers
58 views

Expressing logarithms as a single one: $\log_a (a/\sqrt{x}) - \log_a \sqrt{ax}$

I'm doing problems in my book that say "Express as a single logarithm and, if possible, simplify". There's two I did that I'm not sure about, and they're even numbered unanswered problems. The ...
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2answers
63 views

Proving $\lim_{x\to \infty}\ln(x)/x$

Can you please check if my proofs are correct? for $$\lim_{x\to +\infty}\ln(x)=+\infty$$ I used the mean value theorem : $\ln$ continuous on $[1,x]$ $\ln$ differentiable on $(1,x)$ then there ...
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2answers
47 views

Prove convexity of function over space of positive definite matrices

I want to show that the function $f(X) = -log \ det(X)$ is convex on the space $S$ of positive definite matrices. What I have done: It seems like this problem could be tackled by considering the ...
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2answers
26 views

Deriving difference from difference of logarithms

Good afternoon. I know that $\log{x} - \log{y} = -0.204$. How do I compute $x - y$? Thanks a lot for your solutions!
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2answers
66 views

Given $e=\sqrt[y]{x}$ isolate y

I have a problem trying to create a function in a programming language that does not support any functions other than that of basic arithmetic (addition, subtraction, exponentiation, division...). ...
1
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1answer
34 views

Log of 1-reguralized incomplete gamma function (upper)

I must compute value of 1-regularized incomplete gamma function (upper) $Q(a,z)$. But unfortunatelly this computing exceeding the precision of the processor (for example I gain 0, but I should have ...
17
votes
4answers
2k views

How does e, or the exponential function, relate to rotation?

$e^{i \pi} = -1$. I get why this works from a sum-of-series perspective and from an integration perspective, as in I can evaluate the integrals and find this result. However, I don't understand it ...
2
votes
3answers
60 views

Is it correct to say that $\log_2 0=-\infty$?

The logarithm is not defined at $x=0$, because it tends to $-\infty$ as x tends to 0 from above. But is it nevetheless correct to say that $$ \log_2(0)=-\infty? $$ Or is it better to say/ write $$ ...
1
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2answers
36 views

Find $\log_c{x}$ if $\log_a{x} = p$, $\log_b{x} = q$, and $\log_{abc} {x} = r$.

Given that $\log_a{x} = p$, $\log_b{x} = q$, and $\log_{abc} {x} = r$, find the value of $\log_c{x}$.
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0answers
11 views

MSE in case of log-transformed dependent variable

Let's consider the following log-linear model: $log(Y_i) = \alpha + X_i\beta + \epsilon_i$ for i = 1, ..., N The fitted value is: $\widehat{log(Y)} = \hat{\alpha} + X\hat{\beta}$ Assuming ...
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2answers
25 views

Logarithmic inequality: $\log_{1/3}^2(x^2-3x+2) - \log_{1/3}(x-1)>\log_{1/3}(x-2) +6$

I need help solving this: $$\log_{1/3}^2(x^2-3x+2) - \log_{1/3}(x-1)>\log_{1/3}(x-2) +6$$ So far I could not make sense of this, because I don't understand how to handle $\log^2$ or the $+6$ at ...
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2answers
21 views

Express the given expression as a single logarithm

Express $$2 \ln (2 - x) + 3 \ln (x^2 - 5)$$ as a single logarithm. Can anyone help me with this question? Thanks
0
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1answer
55 views

The limit of $ m\int_{a}^{1/m} \frac{dx}{x}=0 $ and $ m\int_{a}^{\infty} \frac{dx}{x^{1+m}}=0$ as $m\to0$

Given $ a >0 $ is it correct that $$ \lim_{m\to 0}m\int_{a}^{1/m} \frac{dx}{x}=0 $$ by the properties of the logarithm function? Or on the other hand, $$\lim_{m\to 0} m\int_{a}^{\infty} ...
6
votes
1answer
216 views

Contour Integral $ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$

I need help evaluating this with contour integration $$ \int_{0}^{1}{\ln\left(\,x\,\right)\over \,\sqrt{\vphantom{\large A}\,1 - x^{2}\,}}\,{\rm d}x $$ I am not sure as to how to work with the branch ...
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2answers
6k views

What is log ? What does it mean? How does it transform a number?

What is log ? What does it mean? How does it transform a number? /I'm a code and see log being used in legacy code I have to change. Thanks
2
votes
1answer
60 views

Does $\log_a b = \log_\sqrt a \sqrt b$ can be a basic logarithm law?

Does the following equation is true for all $ a,b\in{\mathbb R}$? $$\log_a b = \log_\sqrt a \sqrt b$$ I have tried to proove this, and I didnt find any contradiction. Is it true? EDIT Thanks guys ...
10
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4answers
341 views

How to evaluate $\int_0^1 (\arctan x)^2 \ln(\frac{1+x^2}{2x^2}) dx$

Evaluate $$ \int_{0}^{1} \arctan^{2}\left(\, x\,\right) \ln\left(\, 1 + x^{2} \over 2x^{2}\,\right)\,{\rm d}x $$ I substituted $x \equiv \tan\left(\,\theta\,\right)$ and got $$ ...
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1answer
48 views

Maximal number of colours in a palette that allows for fewer than 500 mixtures

An artist is planning on mixing together any number of different colours from her palette. A mixture results as long as the artist combines at least two colours. If the number of possible mixtures is ...
2
votes
2answers
43 views

What is the limit of $\lim_{x\rightarrow 0} (\log _{\tan^2x}\tan^22x) $

How do i calculate the limit of this function? $$ \lim_{x\rightarrow 0} (\log _{\tan^2x}\tan^22x) $$ I have no idea where to start.
0
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2answers
21 views

comparison test to show that $\sum_{n=1}^{\infty}\frac{1}{(n+2)\sqrt{ \ln ^3(n+3)}}$ converges

As the title says, I know that this sum converges and I want to find a suitable comparison test. Cauchy's root test and d'Alembert's ratio test gave inconclusive results. According to wolfram this ...
0
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1answer
56 views

Indefinite integrals with natural logs [duplicate]

I know the integral of $\frac{1}{x}$ is $\log(x)$ but I'm not sure how to solve this problem, any help would be appreciated: $$ \int^{3}_{2} \frac{1}{x \ln x} $$ I think I need to substitute $x\ln x$ ...
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vote
1answer
41 views

Is there any holomorphic function in a unit ball such that $f(1/n)=\frac{1}{n \ln{n}}$

Is there any holomorphic function in a unit ball such that $f(1/n)=\frac{1}{n \ln{n}}$ for $n=2,3,\dots$ Can somebody point me in the right direction? Only thing I know is that $f(0)=0$ if such ...
0
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1answer
104 views

how can I publish my log approximation formula

I've successfully found out a formula which can give log value of any base till 4-5 places after decimal I want to know whether it can get published because I've seen some journals which have ...