Questions related to real and complex logarithms.

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0answers
16 views

Fisher Transform: Use of the natural logarithm of negative number. How is it possible?

I have the following equation, from http://www.mesasoftware.com/Papers/USING%20THE%20FISHER%20TRANSFORM.pdf Because the parameters inside the log are (1+x)/(1-x), the output is always negative when ...
1
vote
1answer
290 views

logarithmic function between two points

I need to find the logarithmic curve between two points $$A(0,5),\quad B(180,9)$$ We know that the formula for logarithmic function is: $\;f(x) = \log(x)\,\;$so $$ 5 = \log(0),\quad 9 = \log(180)$$ ...
3
votes
2answers
92 views

Find $\frac{1}{\log 2}+\frac{1}{(\log 2)(\log 3)}+\frac{1}{(\log 2)(\log 3)(\log 4)}+ \cdots$

Is it possible to calculate the sum of $\dfrac{1}{\log 2}+\dfrac{1}{(\log 2)(\log 3)}+\dfrac{1}{(\log 2)(\log 3)(\log 4)}+ \cdots$?
12
votes
3answers
300 views

A closed form for $\int_{0}^{\pi/2}\frac{\ln\cos x}{x}\mathrm{d}x$?

The following integrals are classic, initiated by L. Euler. \begin{align} \displaystyle \int_{0}^{\pi/2} x^3 \ln\cos x\:\mathrm{d}x & = -\frac{\pi^4}{64} \ln 2-\frac{3\pi^2}{16} ...
11
votes
4answers
855 views

Evaluating $\int_0^{\large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $

It's my first post here and I was wondering if someone could help me with evaluating the definite integral $$ \int_0^{\Large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $$ Thanks in ...
0
votes
1answer
69 views

solving mod equation

I am attempting to solve $r_1$ in this equation: $$m + xr \equiv m_1 + xr_1 \pmod q$$ This is what I derived at: $$m-m_1 + xr / x \equiv r_1 \pmod q$$ I proceed to sub these with the necessary ...
12
votes
4answers
377 views

Evaluate $\int^1_0 \log^2(1-x) \log^2(x) \, dx$

I have no idea where to even start. WolframAlpha cant compute it either. $$\int^1_0 \log^2(1-x) \log^2(x) \, dx$$ I think it can be done with series, but I am not sure, can someone help a little so ...
3
votes
2answers
104 views

Proof of a closed form of $\int_0^1(-\ln x)^ndx$

$$\int_0^1(-\ln x)^ndx$$ Is there a step-by-step solution to a closed form of this expression? I've tried using different representations to re-write the expression but I couldn't find anything I knew ...
0
votes
2answers
428 views

Quick logarithm calculation

In coming up with an algorithm for finding log (10) base 2, these are my thoughts. I wanted to know if this makes sense and how could I truly make it more efficient. The requirements are strictly not ...
2
votes
2answers
128 views

Solving the equation $\ln(x)=-x$

I tried solving this equation for a long time but did not succeed. Any help is appreciated. $$\ln x=-x$$ I am not sure the tag is correct, I am not familiar with English mathematical terms. Please ...
2
votes
2answers
82 views

Show without derivative that function $\frac{\ln{n}}{ n\ln{\ln{n}}}$ is decreasing

I have a problem with showing the function $\displaystyle \frac{\ln{n}}{n \ln{\ln{n}}}$ is decreasing. I came to form $(n+1)^{\ln{\ln{n}}}<(n)^{\ln{\ln{(n+1)}}}$ and I don't know how to show that ...
2
votes
3answers
126 views

Show without differentiation that $\frac {\ln{n}}{\sqrt{n+1}}$ is decreasing

Show that the function $\displaystyle \frac {\ln{n}}{\sqrt{n+1}}$ is decreasing from some $n_0$ My try: $\displaystyle a_{n+1}=\frac{\ln{(n+1)}}{\sqrt{n+2}}\le ...
2
votes
2answers
56 views

Upperbound this difference between two log expressions

I have the difference between the following log expressions and I am trying to bound the difference, $$F= \log \left(1+ \left(2+\frac{1}{\sqrt{2}}\right)^2 x^2\right) - \log \left(1+ ...
6
votes
1answer
107 views

$\log^2 (x^2) + \log (x-1) = 0$

I'm trying to solve the equation $\log^2 (x^2) + \log (x-1) = 0$ but all I could do is to show that $1 < x < 2$. Wolfram Alpha says that $x = 1.508554...$, this is good, but I really want to ...
0
votes
0answers
35 views
3
votes
8answers
298 views

Solution of an exponential equation

Probably very simple question. Why the solution of $$1=n(1-a)^{t}$$ in terms of $t$ is equal to: $$t=\frac{\ln n}{\ln \frac{1}{1-a}}$$
7
votes
5answers
271 views

Solving the exponential equation: $3 \cdot 2^{2x+2} - 35 \cdot 6^x + 2 \cdot 9^{x+1} = 0$

I have this exponential equation that I don't know how to solve: $3 \cdot 2^{2x+2} - 35 \cdot 6^x + 2 \cdot 9^{x+1} = 0$ with $x \in \mathbb{R}$ I tried to factor out a term, but it does not help. ...
2
votes
3answers
74 views

Finding the limit $\lim_{n\to\infty} \frac{n\left(\sqrt[n]{n}-1\right)}{\log n}$

I try to calculate the following limit: $$\lim_{n\to\infty}\frac{n\left(\sqrt[n]{n}-1\right)}{\log n}$$ I think it should equal 1, because: $$\exp(x)=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^{n}$$ ...
21
votes
1answer
682 views

Integral $\int_0^\infty\frac{\ln\left(\sqrt{x+1\vphantom{x^0}}-1\right)\,\ln\left(\sqrt{x^{-1}+1}+1\right)}{(x+1)^{3/2}}dx$

Another integral similar to my previous question: $$\int_0^\infty\frac{\ln\left(\sqrt{x+1\vphantom{x^0}}-1\right)\,\ln\left(\sqrt{x^{-1}+1}+1\right)}{(x+1)^{3/2}}dx$$ Could you suggets how to evaluate ...
1
vote
1answer
40 views

How do I prove this derivation?

I hope you can help me with this one because I seem to not quiet get a start here :/ Lets say we got a $b\in\mathbb{R}_{\gt 0}$ and a $y\in\mathbb{R}$ and we define $b^y:=\exp\left(\ln b \cdot ...
0
votes
2answers
1k views

Graphing: Given two points on a graph, find the logarithmic function that passes through both.

Is there such a method to do this? I would like to come up with a logarithmic function (a graph that looks like a square root graph) that passes through two given points. Haven't had any luck in ...
1
vote
2answers
69 views

Why does this inequality stand?

I want to ask something about: "Since $i \log_e i$ is concave upwards, it is easy to show that $$\sum_{i=2}^{n-1} i \log_e i \leq \int_2^n x \log_e x \,dx \leq \frac{n^2 \log_e ...
5
votes
1answer
48 views

How to prove that $f(x) - f(x-1)$ approaches $\frac{\log_{10}(10)}{\log_{10}(e)}$?

Let $$f(x) = \sum_{n=1}^{10^x}\frac{1}{n}$$ I noticed that as x approaches $\infty$, $f(x) - f(x - 1) \approx 2.3025$. After a bit of experimenting, I found that $2.3025... = ...
1
vote
2answers
100 views

How does $n < 2^n$ become $\log n < n$ by taking log of both sides?

How does $n < 2^n$ become $\log n < n$ by taking the log of both sides? I understand the left side but I do not understand the right side of the inequality. The once was $\log 2^n$ becomes $n$ ...
2
votes
1answer
41 views

Find the leftmost (most significant digits) of a large exponent calculation, say $99^{99}$

I want to find the initial 10 digits of an exponent calculation whose result is a very large number - Say, $99^{99} = 3.697296 \times 10^{197}$ I only need to know the digits $3697296$ Is there any ...
0
votes
1answer
26 views

Problem on logarithms

if lg2=x, lg3=y ,then i) 2/9 ii) 75 iii) 0.0015 Write logarithm base 10 of x and y Please help me to resolve this problem. For first one I got this, Is this correct? 10^x=2 10^y=3 =10^x / (10^y)^2 ...
7
votes
1answer
73 views

$a_{n+1}=\log(1+a_n),~a_1>0$. Then find $\lim_{n \rightarrow \infty} n \cdot a_n$

Suppose that $a_{n+1}=\log(1+a_n),~a_1>0$. Then find $\lim_{n \rightarrow \infty} n \cdot a_n$. I can find $\lim_{n \rightarrow \infty}a_n=0$. But I have no idea to find $\lim_{n \rightarrow ...
0
votes
1answer
57 views

Reasoning behind multiplying by conjugates

What is the reason behind multiplying by conjugates? I am currently studying single variable calculus and throughout the lessons from the text I'm using, the reasoning as to why one would multiply by ...
1
vote
1answer
23 views

Double solutions and plotting transcendental equations

I have the following transcendental equation: $y^2 - \log(y)^2 = 4\cdot\log(x) + 4/x + C$ and I aim to plot the equation in the positive, real quadrant, with $x>0$ (actually in the $0 < x ...
3
votes
2answers
72 views

How many numbers less than $x$ have a prime factor that is not $2$ or $3$

I am trying to figure out the number of integers greater than $1$ and less than or equal to $x$ that have a prime factor other than $2$ or $3$. For example, there are only two such integer less than ...
2
votes
0answers
34 views

Existence and uniqueness of a function generalizing a finite sum of powers of logarithms

I hope to find a proof of the following conjecture: $(1)$ For every $a>0$ there is a convex analytic function $f_a:\mathbb R^+\to\mathbb R$ such that: $f(1)=0$ and $\forall x>1,\ ...
4
votes
2answers
75 views

How to integrate $\ln \big( b + \sqrt{b^2 + c^2 + x^2}\,\big)$?

I am looking to demonstrate the following result. Any ideas are much appreciated. $$ \begin{align}\int \ln \left( b + \sqrt{b^2 + c^2 + x^2}\right) dx = &\;x \ln \left( b + \sqrt{b^2 +c^2 ...
2
votes
1answer
21 views

Discrete logarithm when $\alpha$ is not a primitve root

When a number $\alpha$ is a primitive root for a prime number $n$ then $\beta \equiv \alpha^{x} \mod n$ can be written as $x = \log_\alpha(\beta) \mod n-1 $. If $n$ is not a prime, the equation ...
3
votes
1answer
50 views

Proving analytic continuation, choosing suitable branch cuts,

Consider the function $$f(z)=\log[(z^2+1)^{1/2}],\quad z>0$$ where the branch is chosen so that $(z^2+1)^{1/2}>0$ for $z>0$ and the log denotes the principal branch. Let $R$ be the union of ...
0
votes
3answers
25 views

Limit and L'Hopitals

I'm having trouble with this problem. $\lim{n \to \infty} (1+\frac 3n)^n$ My professor said to use a proof to figure out that the limit of the ln of the function is 3, but I can't figure out how to ...
5
votes
6answers
894 views

Alternate proof for “$\log_{10}{2}$ is irrational”

I need to prove that $\log_{10}{2}$ is irrational. I understand the way this proof was done using contradiction to show that the even LHS does not equal the odd RHS, but I did it a different way and ...
2
votes
2answers
2k views

Why are logarithms not defined for 0 and negatives?

I can raise $0$ to the power of one, and I would get $0$. Also $-1$ to the power of $3$ would give me $-1$. I think only some logarithms (e.g log to the base $10$) aren't defined for $0$ and ...
0
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1answer
40 views

Interpret a linear scale as a logarithmic scale

Answer The solution to my question is $10^{\log{logMin}+\frac{x-linMin}{linMax-linMin}(\log{logMax}−\log{logMin})}$ with $x$ being a value on a linear scale $linMin$ to $linMax$ that is being ...
1
vote
1answer
185 views

CAGR, log of negative numbers

I am trying to calculate the Compound Annual Growth Rate (CAGR) of a number of companies using the Geometric mean of the annual growth of their EPS. Some of the EPS values are negative(loss making) ...
0
votes
3answers
53 views

Find the derivative of $F(x) = x^7 \ln(x^3 e^{3x^2 -8})$

Find $F'(x)$ for $$F(x) = x^7 \ln(x^3 e^{3x^2 -8})$$ Here is what I have so far: $$(7x^6)(3x^2)$$
1
vote
1answer
49 views

Is it possible to have Logarithm with base 1 or 0?

I am wondering is there any definition that allows logarithm to have base 0 or 1 in real or complex fields (considering Euclidean space)?? Out-coming question is if you can define a logarithm with ...
0
votes
1answer
22 views

Some trouble with algebra using logarithms and summations

I'm having some embarrassing trouble with algebraic manipulation. I have the function $$f(y) = y^Tx-\log\sum_{i=1}^ne^{x_i}$$ and for each $i = 1,2,\ldots,n$ $$y_i = {e^{x_1} \over ...
-1
votes
1answer
24 views

Solve the below equation For $X$ in terms of $Y$ and $Z$ [closed]

Solve the below equation For $X$ in terms of $Y$ and $Z$ $XY-log(X)=Z$ Also give any other method like simulation, graph plot
2
votes
4answers
39 views

How can I differentiate this equation?

I need to differentiate this: $$ y = b(e^{ax}-e^{-ax}) $$ I've got the solution from a book, but I don't found the process to differentiate it. The solution is: $$ y = ab(e^{ax}+e^{-ax}) $$ Here ...
0
votes
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 $\ln(x)$: $\ln(x)= (x - 1) - \frac{(x-1) ^ 2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4} \cdots$ ...
1
vote
2answers
331 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 ...
0
votes
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 ...
0
votes
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
votes
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 ...
0
votes
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 $0.05$.