Questions related to real and complex logarithms.

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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
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
1answer
47 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
37 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' ...
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. ...
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 ...
1
vote
4answers
68 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!
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 ...
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?
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: ...
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 ...
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 ...
2
votes
2answers
41 views

A limit concerning the integral of $x^n$

I pondered if the general integral of $x^n$ could be used with limits to prove that $$\int x^{-1}dx=\ln(x)+C$$ I started with $$\int x^ndx=\frac1{n+1}x^{n+1}+C$$ Then, I took the limit as $n$ ...
0
votes
3answers
37 views

How do i solve these exponential equations? [closed]

Is there a way to solve these exponential equations without using logarithms? I tried to get the same base for all the terms, but I could not make it. Is there any other general procedure that I can ...
1
vote
2answers
64 views

$3^{\sqrt{\log_{3}{x}}}+x^{\sqrt{\log_{3}{x}}}=6\Rightarrow\, x=?$

How can i solve the following equation? $$ 3^{\sqrt{\log_{3}{x}}}+x^{\sqrt{\log_{3}{x}}}=6 $$ It is clear that $x=3$ is a solution of this equation. But how can i prove that there is another solution ...
0
votes
3answers
32 views

Express $y$ in terms of $x$ in logarithmic graph

Express $y$ in terms of $x$: I know that $y = mx + c$ translates to: $\log y = n \log x + \log c$ All I can see in the question 2a of the graph below. I can tell that from the graph in question ...
1
vote
3answers
35 views

Let $x=\frac{1}{3}$ or $x=-15$ satisfies the equation,$\log_8(kx^2+wx+f)=2.$

Let $x=\frac{1}{3}$ or $x=-15$ satisfies the equation,$\log_8(kx^2+wx+f)=2.$If $k,w,f$ are relatively prime positive integers,then find the value of $k+w+f.$ The given equation is ...
5
votes
2answers
115 views

Integral of the following function: [duplicate]

$$I=\int_{0}^{\dfrac\pi4}\log(\cos(x))\mathop{\mathrm{d}x}$$ I can solve it if the limit is from $0$ to $\frac\pi2$. How to do it? I have done like this, tried not to use any knowledge of series but ...
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 ...
4
votes
1answer
67 views

Convergence/divergence of the sum $\sum_{n=2}^\infty 1/ \ln(n!) $

Is the sum $$ \sum_{n =2}^{\infty} \frac{1}{\ln n!} $$ convergent or divergent? I have tried different methods and it doesn't work. Perhaps comparing with a divergent series will work? I'm thinking ...
0
votes
1answer
40 views

Logarithms Problem, when finding $x^n = x$

Why is it that $1^4 = 1$, when using log laws why do you get $3 = \frac {\ln1}{\ln1} = 1 \therefore 2 = 0$? I was trying to show that $(-1)^{2^x + 1} = -1$, given $x \geq 0 $ and is an integer, but ...
1
vote
2answers
57 views

Why $n^{\log_2 n}$ almost equals $2^{\log^2_2 n}$?

Just by doing some calculations: ...
2
votes
1answer
32 views

Tight lower bound for logarithm function

Is there a lower bound for the logarithm function which is tighter than, $$\log(x)\geq 1-x^{-1}$$ that works for all real values of $x>0$?
-1
votes
1answer
17 views

Prove that Log is defined on D [closed]

$D=D(0,R)$ is the disk of center $0$ and radius $R$. Given that $a>R$ and $\Phi(z)=\frac{a-z}{a+z}$, I have proved that $\forall z\in D$, $\operatorname*{Re}(\Phi(z))>0$. Prove that $f = ...
0
votes
1answer
41 views

Show that $y=e^{e^{cx}}$ is a solution of the differential equation $\frac{d^2y}{dx^2} =c^2 \cdot y \cdot \ln(y) (1+\ln(y))$

Question: Show that $y=e^{e^{cx}}$ is a solution of the differential equation $$\frac{d^2y}{dx^2} =c^2 \cdot y \cdot \ln(y) (1+\ln(y))$$ I know there are a lot of ways of solving this ...
0
votes
4answers
95 views

Prove $\log(n!) =\Omega(n\log(n))$ [closed]

Can someone help me prove that $\log(n!) =\Omega(n\log(n))$, that is, that there exists some positive $c$ such that, for every $n$ large enough, $\log (n!)\geqslant c\cdot n\cdot \log(n)$?
0
votes
2answers
47 views

Rules of logarithm

Can anyone help me figure out how to go from the first expression to the second? $$ \begin{equation} \ln D=u+\delta(e-p)+\gamma y-\sigma r \end{equation} $$ $$ \begin{equation} \pi \ \ln (D/Y)= ...
0
votes
2answers
30 views

For what values are these logarithms true?

For what values, x and y, are both these equations true? $$\frac {\log(x)}{\log(y)} = \frac 23$$ AND $$\frac xy = \frac 23$$ How would one solve this?
1
vote
0answers
61 views

Natural Logarithm Integration

For what set of functions is $\int \frac{\ln{f(x)}}{f'(x)}\mathrm{d}x$ defined? More specifically related to the reason I ask, when $$ f(x) = \frac{c}{x} + \arcsin{x} + \sqrt{\frac{1}{x^2}-1} $$ is ...
0
votes
1answer
29 views

Solving an expression containing two added exponential functions

I have a problem solving the below equation with respect to $x$: $0.6\cdot \exp(\frac{-40}{x})+0.4 \cdot \exp(\frac{10}{x})=1$ My problem is that I have two exponential functions which are added ...
3
votes
2answers
37 views

logarithm equation with scalar on right hand side

$\log_7 (x^2-1) - \log_7 (x-1) = 2$ $\log_7 49 = 2$ => $\log_7 (x^2-1) - \log_7 (x-1) = \log_7 49$ => $\frac{(x^2-1)}{x-1} = 49$ => $x^2 -1 = 49(x -1)$ => $x^2 -1 = 49x -49$ => $x^2 - 49x + 48 = ...
2
votes
1answer
44 views

I am approximating $\ln x$ and $\log x$. How could I make these curves into a general equation?

Because I am waiting for my graphing calculator to ship, I need a quick-and-dirty way to calculate logarithms on a four-function calculator (for when I need to keep my laptop away from where I work). ...
1
vote
3answers
44 views

Derive $\log(a+b)=\log(a)-2\log\left(\cos\left(\arctan\left(\sqrt{\frac{b}{a}}\right)\right)\right)$

In the comments section of another post, MATHEMATIKER stated that $$\log(a+b)=\log(a)-2\log\left(\cos\left(\arctan\left(\sqrt{\frac{b}{a}}\right)\right)\right)$$ if $b>a>0$. I wish to know ...
1
vote
2answers
55 views

Find $b$ such that $\log_b(x)$ and $\log_b(y)$ are integers.

Is it possible to find a value $b$ such that, when given $x,y\in\mathbb{N}$, $\log_b(x)$ and $\log_b(y)$ result in integers? My assumption is that if $b\in\mathbb{Z}$, then $b$ may not exist, but ...
0
votes
3answers
46 views

how tell if a series of power numbers is bigger then others

I trying to order a list of mathematical expressions in string format as: "2*2" "4^1" "4^2^5" so far, so good for non exponential operations (^). I could compute ...
1
vote
2answers
82 views

Limit of the sequence $\frac{1}{n}\left[\log\left(\frac{n+1}{n}\right)+\log\left(\frac{n+2}{n}\right)+\dots+\log\left(\frac{n+n}{n}\right)\right]$

How can we evaluate the following limit $$ \lim_{n\to\infty}\frac{1}{n}\left[\log\left(\frac{n+1}{n}\right)+\log\left(\frac{n+2}{n}\right)+\dots+\log\left(\frac{n+n}{n}\right)\right] $$
10
votes
3answers
403 views

Number system with $e^x = 0$ for some $x$

It is well known that $e^x \ne 0$ for all $x \in \mathbb{R}$ as well as $x \in \mathbb{C}$. Upon reading this article and doing a bit of research I have found that this also applies to the ...
1
vote
1answer
73 views

How I can decompose $\ln(3f(x)+2g(y))$

I'm trying to simplify this equation: $$\ln(3f(x)+2g(y))$$ where $f$ is a function like $f=2x$ and $g$ is another function like $g=x²$ Can I rewrite this equation? Any help will be appreciated! ...
0
votes
4answers
71 views

Approximation $\log_2(x)$

Can anyone share an easy way to approximate $\log_2(x)$, given $x$ is between $0$ and 1? I'm trying to solve this using an old fashioned calculator (i.e. no logs) Thanks! EDIT: I realize that I ...
1
vote
2answers
37 views

How to find the unknown in this log inequality??

Find all values of the parameter a $\in\Bbb R$ for which the following inequality is valid for all x $\in\Bbb R$. $$ 1+\log_5(x^2+1)\ge \log_5(ax^2+4x+a) $$ I'm lost when I got to this stage: $ ...
3
votes
2answers
41 views

Finding the limit as $n \to \infty $ of $n\ln\left(1+\frac{\ x}{n^2}\right)$

Find $$\lim_{n\to \infty} n\ln\left(1+\frac{\ x}{n^2}\right)$$ My attempt: $\lim_{n\to \infty} n \left[\ln\left(\frac{\ n^2 +x}{n^2}\right)\right]$ = $\lim_{n\to \infty} n [\ln (n^2 +x) - ...
0
votes
1answer
20 views

Solve equation with variable in the fraction of a logarithm

I really had a hard morning thinking about how to solve an equation for a variable while the variable we want to solve for is in the fraction of a natural algorithm. I have this particular equation: ...
3
votes
1answer
69 views

Do those iterated increment rates always yield monotonic functions?

For any $a\in{\mathbb R}$ and any non-empty open interval $I$ containing $a$, we have an operator $T_a$ on $C^{\infty}(I,{\mathbb R})$ defined by $$ T_a(f)(x)=\left\lbrace \begin{array}{lcl} ...
1
vote
1answer
36 views

how to solve an nth derivative for the equation $\ln((1+x)/(1-x))$

I'm trying to find the $n$th derivative of this function. I've got that the first term is: $$ \frac{2(n!)x^{n-1}}{(x^2-1)^n} $$ Any improvement on this would be very helpful.
2
votes
1answer
46 views

What's wrong with my infinite series expansion for $\log(x)$?

Here, log is natural log. Looking at $f(x)=\frac{1}{x}$, I tried to put $f(x)$ in the form $\frac{a}{1-r}$ that an infinite geometric series $\sum_{n=0}^\infty (a \cdot r^n)$ converges to when $\mid ...
-6
votes
2answers
43 views

How to solve $\log(a^b)=b$? [closed]

How to solve for $b$ when log a common log: $$\log a^b=b$$ Please, denote the solution step-by-step. Any property denotation will also be very useful.
0
votes
1answer
34 views

How do I prove $\sum_{i=0}^{\log_3{n}}3^i = \frac{3n - 1}{2}$?

I started my data structures course at university and I came across with that equation, can someone explain me how I prove it please? $$\sum_{i=0}^{\log_3{n}}3^i = \frac{3n - 1}{2}$$ $$3^0+3^1+ ...