Questions related to real and complex logarithms.

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-5
votes
1answer
57 views

Proof that $0^0 \neq 1$ [closed]

Suppose that $t = \sqrt{t}^{\sqrt{t}}$, then, it follows that; $$ t^{\sqrt{t}} = \sqrt{t}^{t} \\ \frac{1}{2}t\ln{\left(t\right)} = \sqrt{t}\ln{\left(t\right)} \\ \ln{\left(t\right)}\left[\frac{1}{2}t ...
4
votes
6answers
3k views

Is there an approximation to the natural log function at large values?

At small values close to $x=1$, you can use taylor expansion for $\ln x$: $$ \ln x = (x-1) - \frac{1}{2}(x-1)^2 + ....$$ Is there any valid expansion or approximation for large values (or at ...
1
vote
2answers
83 views

How to solve the equation $x \log \log x = n$

I would like to solve the equation $x \log\log x = n$. I've seen a lot of post about the equation $x \log x$ but here I have a composition of $\log$. How can I solve it ? Thank you very much.
-1
votes
0answers
10 views

Calculating interest rate with knowing fv, pv, nper, and pmt

I need to create a formula whereby the formula finds the interest rate (R) by input of nper, pmt, pv and fv. I have this question below: You need 50,000 to buy a used Jaguar, but you only have $5,000 ...
0
votes
1answer
26 views

basic math problem relating to log thought my answer is not matching with the alternative

$1/2 \log c = 0.915$. Calculate $c$. It is a basic math problem but my answers are not matching with the alternatives. $1/2 \log c= c^{1/2}$ $ c^{1/2}= 0.915$ $c = 0.915 \times 0.915=0.83448$ ...
1
vote
1answer
13 views

Verify whether or not expression is true or not for z>0

I need to verify whether or not the below expression is true or not for $z>0$. I'm trying to understand the rules of logarithms but I can't figure out how to apply it myself or where to even begin. ...
1
vote
2answers
52 views

Solution of $5^{\log x}+5x^{\log 5}=3$

Solve for $x$ $$5^{\log x}+5x^{\log 5}=3$$ where base of log is $a$, $a>0$ and $a \neq1$ Could someone hint as how to initiate this question? I am not having any idea as how to proceed.
2
votes
1answer
71 views

How to prove $({\log_2 x})^{n+1} \le x^n$

I want to show that $({\log_2 x})^{n+1} \le x^n$ when $n \ge 1$ and $x \ge 1$. I know that ${\log_2 x}$ can be shown to be $\lt x$ with: $x \lt 2^x$ $\log_2 x \lt x$ and obviously adding the same ...
1
vote
1answer
28 views

How are floating-point numbers logarithmically distributed?

From what I remember from a lecture I had of a course I'm attending called "introduction to computational science", floating-point numbers are distributed logarithmically. What does it mean? And how ...
1
vote
2answers
113 views

Does the series $\sum\limits_{n=1}^{\infty}\dfrac{\ln(\sqrt{n})}{n}$ converge or diverge?

How to determine does this series converge or diverge? I have tried d'Alembert's ratio test but in the limit I get $1$. I suppose I should compare it with some other series, but I can't figure out ...
6
votes
5answers
463 views

Evaluate $\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$ using complex analysis

How do I compute $$\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$$ What I am doing is take $$f(z)=\frac{(\log z)^2}{1+z^2}$$ and calculating $\text{Res}(f,z=i) = \dfrac{d}{dz} \dfrac{(\log ...
7
votes
10answers
396 views

How to prove that $\log(x)<x$ when $x>1$?

It's very basic but I'm having trouble to find a way to prove this inequality $\log(x)<x$ when $x>1$ ($\log(x)$ is the natural logarithm) I can think about the two graphs but I can't find ...
11
votes
2answers
173 views

What is $\int_0^1 \ln (1-x) \ln \left(\ln \left(\frac{1}{x}\right)\right) \, dx$?

There are well-known closed-form evaluations for integrals of the form $\int_0^1 a(x) \ln \left(\ln \left(\frac{1}{x}\right)\right) \, dx $ for certain algebraic functions $a(x)$. For example, an ...
0
votes
1answer
615 views

What is the limit of difference between harmonic series and natural logarithm of n+1?

I'm an undergraduate student in geology and I'm dealing with a project in math. The last question of the project gives me the harmonic series ($A_n = 1 + \frac{1}{2} + ... + \frac{1}{n}$) and this ...
1
vote
0answers
27 views

Nested logarithms to represent real intervals.

Out of curiosity from this question regarding writing real numbers as an iterated sum/difference of square roots I started experimenting with another family of functions: $$\log[ \exp[1] \pm ...
0
votes
1answer
31 views

When will the population of a sample double (using dif-eq)?

I have the initial equation $$\frac{dP}{dt}=kp$$ where P is the population, t is time, and k is some positive constant. The rest of what I'm given is that P(0) = A, what is the time for the population ...
0
votes
2answers
14 views

Solution to initial condition problem

$y=-ln(1-e^{(t+c)})$ I'm trying to find the solution to the initial condition $y(0)=-ln2$ Isolate c $0=ln(2)-ln(1-e^c)$ $0=ln({2\over1-e^c})$ $-e^c=2-1$ $e^c=-1$ $c=0$ I can't figure out ...
0
votes
2answers
53 views

Tricky Logarithmic inequality

I have tried proving this logarithmic inequality but I did not succeed. I tried to put every term on one side, I expanded and tried to use one of the properties of logarithms but the proof does not ...
0
votes
0answers
15 views

Weighted logarithmic ranking

I want to have a ranking of players by percentage of shots made, weighted by the total number of shots attempted. The weighting should follow a log scale, so for example Player A has 100% accuracy, ...
0
votes
1answer
26 views

What is the argument of the logarithm operator called?

In the expression $ln(y)$, what is '$y$' called. I'm asking for a noun analogous to exponent in $x^n$, where '$n$' is called the exponent. If I'm not mistaken, '$x$' in this case is the radix, or ...
1
vote
1answer
15 views

Is it possible to clear the x using the Lambert function?

$ y = \frac{x^2}{4} - \frac{ln(x)}{2} $ Solving, I get to: $ e^{4y} = \frac{e^{x^2}}{x^2} $ But I don't know how to continue.
0
votes
1answer
46 views

Stuck solving $\ln(e^y-1)-y=t+c$ for $y$

I'm trying to solve for $y$ $\ln(e^y-1)-y=t+c$ $e^y-1=e^{(t+c+y)}$ $e^y=e^{(t+c+y)}+1$ $y=t+c+y+1$ Where am I going wrong?
6
votes
0answers
159 views

Are known these identities, that I've deduce using Mobius inversion formula?

I would to know if this formula is right and know (these formula are the same by exponentiation), since I deduce this easily by a standar way (perhaps there are mistakes) using Mobius inversion from ...
4
votes
2answers
344 views

If $x$ is rational, can $\log(1-x)/\log x$ be algebraic?

If $x$ is positive rational number less than $\frac{1}{2}$, can the following logarithmic expression be equivalent to real algebraic number, say $g$? $$\frac{\log(1-x)}{\log x} = g$$
1
vote
0answers
43 views

Question about the connection between exponential and logarithmic functions

Does this make sense to anyone? What advice would you give me to clarify my reasoning and explanation? One of the really "neat" features of the exponential function: $$f(x)=e^x$$ is the fact that ...
0
votes
1answer
24 views

Is it possible to simplify $y=100x\cdot\log_{x+1} 2$ (Solved)

Is there any way to simplify the following equation, or any way to reconfigure it in a way that is possible to graph? $$y=100x\cdot \log_{x+1} 2$$
0
votes
0answers
13 views

Calculating Split Info with given equation not matching solution

I'm given the formula to calculate the Split info but cannot seem to calculate the correct answer of 0.926 that the example shows. Split Info $= -\Sigma \frac{\mid D_j\mid}{\mid D\mid} * log_2 ...
1
vote
1answer
56 views

How to solve $y= x \cdot 2^x$ for x

Can anyone help me solve this for $x$: $y= x \cdot 2^x$ I know for $y= 2^x$, that $\log_2(y) = x$ And I can get $\displaystyle \frac yx = 2^x \implies \log_2 \frac yx = x $ But I can't ...
0
votes
1answer
21 views

Decomposition of the entropy

So, I'm reading about this property in the MacKay book. But I don't fully get it. Can someone explain it to me? There's this example: A source produces a character $x$ from the alphabet $A = \{0, ...
0
votes
0answers
25 views

Resolving Zeros in Product of items in list.

Given the formula: $\sqrt [ 1/N ]{ \prod _{ n=1 }^{ N }{ { P }_{ n } } } $ where ${ P }_{ n }$ is a list of real numbers, e.g. [0.4, 0.3, 0.2, 0.1] And the ...
0
votes
2answers
19 views

Computing first k digits and last k digits of a large number using logarithm

How do we compute the first $k$ digits and last $k$ digits of a large number say $2^{N-1}$ for bigger values of $N$ using logarithms? An example for the algorithm will be greatly appreciated. I got ...
0
votes
1answer
71 views

does this equation has an answer?

This is the equation:$$x=\log(a+bx),$$ where $a$ and $b$ satisfies the conditions that let the equation makes sense. Does it have an answer that can be expressed explicitly? Thanks a lot.
1
vote
0answers
13 views

Deformations in Variational Bayesian method

I'm studying Topic Model, but I can't understand the following transformations. $F$ is variational lower bound. $$\begin{eqnarray} F[q(z_{1:n}, \phi, \pi)] &=& \int \sum_{z_{1:n}} q(z_{1:n}) ...
3
votes
1answer
74 views

$x+\ln(x)=0$, what is $x$?

My friend came across this strange equation and I cant find mathematical way to find $x$ without drawing $x$ and $-\ln(x)$ and see that they come across at almost $x=0.5$. Can any one help?
2
votes
1answer
28 views

Solving Weird Logarithms without a Calculator

Given "$x = \log 8$", it is very easy to rewrite the expression as "$10^x = 8$", which cannot easily be solved for by hand. However, if I plug "$x = \log 8$" into my calculator, I get "$x = ...
0
votes
1answer
50 views

Domain of $\ln\left(\frac{6}{6+x-x^2}-1\right)+\arcsin\left(\frac{x+1}{3}\right)$

blob:https%3A//mail.google.com/ea67134d-45a0-4cc0-9ec7-abf6d5a50852 I believe that my first condition is wrong but I don't understand why. Can somebody please help?
0
votes
2answers
14 views

find $x$ from logarithm expression for $\log_{10}$ fraction

$$\log_{10} x = 0.5$$ I know if $\log_{10} x = 2$ then $x$ is $100$ but I don't know how to work out for a non obvious answer.
0
votes
2answers
39 views

Can $\ln|\cos x|$ be written as $-\ln|\sec x|$? absolute function

$\ln|\cos x| = \ln|1/\sec x| = \ln|(\sec x)^{-1}|=-\ln|\sec x|$ Is what I am doing valid? Or is it not correct because of the absolute function?
0
votes
2answers
69 views

$\log^2n$, $\log n^2$, $\log \log n$, $(\log n)^2$; What are the differences? [closed]

What is the difference between the following: $\log^2n$, $\log n^2$, $\log \log n$, $(\log n)^2$
3
votes
6answers
97 views

Why isn't $-2$ solution for $x$?

I came across an logarithm problem recently. I don't know why solution to this problem cannot be $-2$. Now, don't downvote now because you don't know why I'm asking this. I know that logarithms' ...
14
votes
3answers
2k views

What is the difference between the three types of logarithms? [closed]

In complex analysis I came across three types of logarithms namely $\ln$, $\log$ and $\text{Log}$. What is the difference between the three?
0
votes
1answer
21 views

Argument principle and the principle branch of the complex logarithm

I've just been reading about complex analysis and came across the Cauchy argument principle. In my understanding you are taking the contour integral of $\frac{f'(z)}{f(z)}$ around a designated path. ...
1
vote
4answers
70 views

relation betwn ln and e

If $f(x) = ln(x)$ and $f^-1(x) = e^x$ then is $e^x = 1/ln(x)$??? because I see $e^9 = 8103$ but $1/ln9 = .455$ How are they reverse? I don't understand!
3
votes
2answers
97 views

Solve $\sqrt x = 1 + \ln(3 + x)$ algebraically

I am having trouble with this homework problem. I am able to graph and find the solution, but I am curious as to how one would do this algebraically. The way I began, was subtracting $1$ on both ...
2
votes
2answers
64 views

What is $\log(0/x)$?

$\log(a/b) = \log(a) - \log(b)$; Is $\log(0/x) = -\log(x)$? I watched a video claiming $\log(1/x) = -\log(x)$, which I get because $1/x = x^{-1}$ and $\log(x^y) = y(\log(x))$ but $\log(1)$... I ...
0
votes
2answers
27 views

Log value to absolute

I am confused how to convert from log value to absolute value from the graph. Below is an example: In the graph, it shows the correlation of age of week and weight of placenta (in log). I can get ...
2
votes
1answer
66 views

How can you solve for s in this very complex problem?

I recently stumped across a problem, which I need to solve. Of course, I used an calculator and I got $s=3$, but I want to know how to do it step by step. The problem is kind of complex: ...
0
votes
1answer
27 views

Numerical derivative of function wrt natural log of variable (non-analytic)

The function that I am trying to evaluate is $$ \frac{d y }{d \ln(x)} $$ where $d$ is the derivative. However I have a set of data points for $x$ and $y$ with uncertainties. Now I think that this ...
0
votes
0answers
24 views

Evaluating right hand limit for a function

I want to prove the following right-hand limit (one sided limit) using $\epsilon-\delta$ definition; $\lim_{u\to 0^+} {u^{s_0} f(-ln u)} = 0$ where $f$ is a function from $R \to R$ and $s_0$ is a ...