Questions related to real and complex logarithms.

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2
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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
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! ...
1
vote
2answers
36 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: $ ...
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 ...
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 $$ \lim_{n\to\infty}\frac{1}{n}\left[\log\left(\frac{n+1}{n}\right)+\log\left(\frac{n+2}{n}\right)+\dots+\log\left(\frac{n+n}{n}\right)\right] $$
2
votes
1answer
313 views

Why is $e$ close to $H_8$, closer to $H_8\left(1+\frac{1}{80^2}\right)$ and even closer to $\gamma+log\left(\frac{17}{2}\right) +\frac{1}{10^3}$?

The eighth harmonic number happens to be close to $e$. $$e\approx2.71(8)$$ $$H_8=\sum_{k=1}^8 \frac{1}{k}=\frac{761}{280}\approx2.71(7)$$ This leads to the almost-integer ...
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 ...
2
votes
1answer
38 views

Find equation from one point and limit

I was wondering if it was possible to find/approximate an equation from just a single known point and the known limit. For example, $$f(0) = 0.8$$ $$\lim_{x \to \infty} \hspace{.1cm}f(x) = 1.8$$ ...
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: ...
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.
-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.
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 ...
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+ ...
0
votes
0answers
22 views

Convert log percentage to linear percentage

I am being provided some numbers in computer code between 0 and 1 that represent a percentage of a base 10 log scale that I need converted to a linear percentage. For example the 50% number for the ...
0
votes
1answer
34 views

Logarithmic to linear

Given this function: $$\frac{1.0}{1024.0} + \frac{x}{100.0} * \frac{1023.0}{1024.0} = y$$ $$10 * \frac{\log_{10}(y)}{\log_{10}(2)} = z$$ $$z * 100 = a$$ ...
3
votes
1answer
96 views

integration of $\ln \ln x$

I would like to compute the following integral : $$\int_{2}^{\frac{\ln a}{\ln \ln a}} \ln \ln x \, \mathrm{d}x$$ where $a$ is a positive constant. Is this possible ?
0
votes
1answer
50 views

$\int_0^5 \frac{dx}{x^2-x-2}$

I am having some difficulty with this problem. I am getting a finite answer but when I put the equation into wolfram alpha to check my answer it says that the integral does not converge.Here is what I ...
0
votes
3answers
63 views

Approximating the value of a definite integral

I came across this question in ISI(Indian Statistical Institute) admission test $$I=\int_2^3 \frac{dx}{\ln(x)} $$ The four options were (A) is less than $2$ (B) is equal to $2$ (C) lies in the ...
0
votes
1answer
37 views

What is the minimum degree of x so that it is greater than or equal to ln(x)?

I was thinking of this question and couldn't find it anywhere. I was trying to find a solution by finding the maximum of the function $f(x) = \frac{ln(ln(x))}{ln(x)}$, yet I'm not sure if that's ...
2
votes
3answers
61 views

Intersection point of two functions - one linear, the other with logarithmic and sqrt terms

I would like first to appreciate everything that is being done on this forum and to greet you all! I have namely two functions and the goal is to find the intersection point of them. $y_1 = a + ...
0
votes
2answers
28 views

How to compute $=\lim_{n \to \infty} \Big( \frac{\log{(n+1)}}{\log{(n)}} \cdot \frac{n-2}{n-1} \Big)$“by hand”?

The problem I'm having is with the logs. I go: $$\lim_{n \to \infty} \Big( \frac{\log{(n+1)}}{\log{(n)}} \cdot \frac{n-2}{n-1} \Big)$$ $$=\lim_{n \to \infty} \Big( \frac{\log{(n+1)}}{\log{(n)}}\Big) ...
2
votes
2answers
76 views

Methods to integrate $(\ln x)^2 $ [closed]

What are some methods to evaluate the integral $$\int \left( \ln x \right)^{2} \, dx \hspace{3mm} ?$$
0
votes
2answers
45 views

Sequence solutions of $ax=e^x$

This question comes from my answer to: Solving $4x = e^x$ without graphing and looking for intersection Here I've used a sequence of nested exponentials constructed from $$ x=\frac{1}{a}e^x $$ and a ...
3
votes
1answer
31 views

How is the principal branch of logarithm defined?

In my textbook, it is defined as: $$\operatorname{Log} z = \ln |z| + i \operatorname{Arg} z$$ Where $\operatorname{Arg}$ is the principal branch of $\arg$, that's, the function which outputs the ...
1
vote
2answers
65 views

Stuck on definite integral problem due to inappropriate $\log$

I have this definite integral problem which I have solved correctly but I'm stuck in one of the steps. I have manipulated it but I think it's not feasible to solve it that way. $$\int_0^a(a^2 + ...
2
votes
1answer
50 views

Stuck on this definite integral problem

I'm stuck on this definite integral problem. I need some constructive hint to proceed further. $$\int_0^a (a^2 + x^2)^\frac{5}{2} dx$$ Substituting $$x = a \cot\theta,$$ I have converted this ...
0
votes
0answers
53 views

Why is numerical integration not working well on logarithm function with bounds $[-1,1]$

When I try to integrate function $x(\log(x)-1)$ from $-1$ to $1$, analytically I get $0.0000 - 1.5708i$ When I try to integrate it numerically, using $10$ points gaussian quadrature I get $0.0000 - ...
0
votes
2answers
35 views

Growth of debt: exponential, logarithmic, or linear? [closed]

If I have increasing debt that I don't intent to pay off for a really long time, how would I prefer to have it grow? Exponentially, logarithmically, or linearly?
0
votes
3answers
42 views

How to take log on this expression

I am solving exact differential equation, but I am stuck on the step on how to simplify this term or how to rewrite it. $e^{-2\ln{\sin{x}}}$
3
votes
1answer
38 views

Why is the discrete log problem intractable?

I have read the other questions on SE on this subject and they were not helpful to me, partially because I am not familiar with advanced mathematical notation. Let me explain the way I'm thinking of ...
2
votes
0answers
24 views

An Integral Substitution for $\int_0^{1} dy \left(\frac{M^2(y)}{\mu^2}\right)^{-\epsilon}$

I have integral (1) as a result from an advanced QFT problem. $$ \tag{1} I= \frac{\alpha}{2\epsilon} \int_0^1 dy \left( \frac{M^2}{\mu^2} \right)^{-\epsilon} + \mathcal{O}(\epsilon) $$ ...
0
votes
0answers
23 views

logarithmic inequality with different bases and root

I have a problem with solving logarithmic inequality $$\log _{\frac{1}{5}}\left(\sqrt{x^3+x^2+x-14}\right)\cdot \log _{\frac{1}{4}}\left(-x^2+5x-6\right)<0$$ My attempt: The domain is ...
0
votes
2answers
33 views

If $x > y$, can you prove $x \log y > y \log x$, $x \ge 1$ and $y \ge 1$

If $x > y$, can you prove $x \ \log y > y \log x$, where $x \ge 1$ and $y \ge 1$. I encountered this problem in a paper I read and somehow cannot prove it.
1
vote
1answer
137 views

Inversion of the function $ \sqrt x \ln x $

Is there an exact (not asymptotic) inversion of the function $ \sqrt x \ln x $ or can we only obtain this inverse in terms of a power series?
0
votes
1answer
18 views

Population decline.

I'm looking at a question here and I'm a bit confused on how I'm supposed to solve it. A population of 460 decreases at 5% monthly. How many years will it take for there to be 100 left on the island? ...
2
votes
5answers
53 views

Solving a three-part log equation using the log laws

The question asks: Solve $$\log_5(x-1) + \log_5(x-2) - \log_5(x+6)= 0 $$ I know that according to log laws, addition with the same base is equal to multiplication and subtraction is equal to ...
0
votes
2answers
26 views

Getting rid off the logarithms in an equation to simplify

ok, I'm having trouble solving for equations when logarithms are involved. I know a little bit about logarithm rules but in equations I'm lost. example: $$\frac{1}{b}\ln{y}=\frac{1}{a}\ln{x}+c$$ I ...
0
votes
1answer
41 views

Bounding a sum of logarithms

Consider a function $f:(0,\infty)\rightarrow \mathbb{N}$ with argument $\epsilon$. Suppose $f$ is decreasing in $\epsilon$. Let $0<b<1$, $K>0$, $d \in \mathbb{N}$, $\delta>0$. Assume $$ ...
1
vote
2answers
55 views

Why is $x^2\ln\sqrt{x}$ equal to $\frac{x^2}{2}\ln x$?

My textbook jumps from $$x^2\ln\sqrt{x}$$ to $$\frac{x^2}{2}\ln x$$ What intermediate steps occur?
26
votes
3answers
2k views

Integral $\int_0^\infty\frac{1}{x\,\sqrt{2}+\sqrt{2\,x^2+1}}\cdot\frac{\log x}{\sqrt{x^2+1}}\mathrm dx$

I need your assistance with evaluating the integral $$\int_0^\infty\frac{1}{x\,\sqrt{2}+\sqrt{2\,x^2+1}}\cdot\frac{\log x}{\sqrt{x^2+1}}dx$$ I tried manual integration by parts, but it seemed to only ...
0
votes
1answer
27 views

how to find the inside value of logarithm?

I m doing sums in chemistry of first order reaction. In it, 0.521 = log(0.3/C) Then how to find the value of c?? The value is c= 0.09
1
vote
2answers
19 views

Exponential decay involving logarithm [closed]

In 2011 reactor $X$ released $4.2$ times the amount of cesium-137 as was leaked during reactor $Y$ disaster in 1986? Using; A = Pert Half-life = $30.2$ years. a) What year will ...
0
votes
2answers
43 views

Inverse of the function $- \log(1-[1-e^{-x^\alpha}]^\beta)$

I have a function as follows, I would like to get the inverse of this function. What is the inverse of $f(x)$? $$ y = f(x) = - \log(1-[1-e^{-x^\alpha}]^\beta)$$ Is my answer correct? $$ f^{-1}(x) = ...
1
vote
2answers
57 views

How do I calculate the inverse function of this function?

I have this function: $$ f(x)=\frac{1+\ln(x)}{1-\ln(x)} $$ And i should calculate $f^{-1}(x)$ I am not really sure how to proceed but I think that the first step would be to have x alone, how do I ...
4
votes
1answer
40 views

Finding the intersections between $y = e^x$ and $y = x + 2$ algebraically?

In trying to find the intersections between $y = e^x$ and $y = x + 2$ in terms of $x$, I came up with the equation, $e^x = x + 2$ and subsequently, $x = ln(x+2)$. Beyond that point, I am stumped. ...
2
votes
3answers
67 views

The growth rate of $(\ln(x))^n$ is a lot slower than I expected

Obviously, the growth rate of $(\ln(x))^a$ is less than the growth rate of $(\ln(x))^b$ as long as $a>b$. Also, the growth rate of $(\ln(x))^n$ is apparently always less than the growth rate of ...
3
votes
1answer
43 views

Summation of $\sum_{k=1}^{n}\left \lfloor \log _{m}k \right \rfloor$ and $\sum_{k=1}^{n}\left \lceil \log_{m}k\right \rceil$ [closed]

$$\sum_{k=1}^{n}\left \lfloor \log _{m}k \right \rfloor$$ $$\sum_{k=1}^{n}\left \lceil \log_{m}k\right \rceil$$ I found myself stuck trying to solve these two summations but i can't make any ...
1
vote
1answer
22 views

Use of asymptotically equivalent equations in limits

I was wondering about the steps to show that the following limit does not exists. $$\lim_{x\rightarrow\infty}[\log(x^2-3)-\log(x+2)]$$ I know that by using L'Hopital's Rule and the continuity of ...
2
votes
4answers
62 views

Solve for $x$ if $4^{\frac{x}{y} + \frac{y}{x}}$ $=$ $32$ and $\log_3(x+y)+\log_3(x-y)=1$

Question: Solve for $x$ if $4^{\frac{x}{y} + \frac{y}{x}}$ $= 32$ and $\log_3(x+y)+\log_3(x-y)=1$ My attempt: With the first equation $$4^{\frac{x}{y} + \frac{y}{x}} = 32$$ ...