Questions related to real and complex logarithms.

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7
votes
8answers
181 views

Calculate $\ln 97$ and $\log_{10} 97$

Calculate $\ln 97$ and $\log_{10} 97$ without calculator accurate up to $2$ decimal places. I have rote some value of logs of prime numbers up to $11$. $97$ is a little big. In case it would ...
4
votes
1answer
103 views

If $\frac{x-1}{e^x-1} = y$ then $x=?$

I have following equation: $$\frac{x-1}{e^x-1} = y$$ I want to solve this equation such that I have the value of $x$ in the term of $y.$ i.e. inverse of the equation
12
votes
0answers
200 views

Relations connecting values of the polylogarithm $\operatorname{Li}_n$ at rational points

The polylogarithm is defined by the series $$\operatorname{Li}_n(x)=\sum_{k=1}^\infty\frac{x^k}{k^n}.$$ There are relations connecting values of the polylogarithm at certain rational points in the ...
11
votes
0answers
121 views

A conjectured identity for tetralogarithms $\operatorname{Li}_4$

I experimentally discovered (using PSLQ) the following conjectured tetralogarithm identity: $$\begin{align}&\phantom{+\;}19\!\;\pi^4-570\ln^42-90\ln^43\\ ...
12
votes
2answers
211 views

Integral $\int_0^1\frac{\log(x)\log(1+x)}{\sqrt{1-x}}\,dx$

I'm trying to evaluate this definite integral: $$\int_0^1\frac{\log(x) \log(1+x)}{\sqrt{1-x}} dx$$ It's clear that the result can be expressed in terms of derivatives of a hypergeometric function with ...
3
votes
4answers
65 views

Evaluating the limit: $\lim_{x\rightarrow -1^+}\sqrt[3]{x+1}\ln(x+1)$

I need to solve this question: $$\lim_{x\rightarrow -1^+}\sqrt[3]{x+1}\ln(x+1)$$ I tried the graphical method and observed that the graph was approaching $0$ as $x$ approached $-1$ but I need to know ...
4
votes
2answers
53 views

Why are values greater than $\pi$ radians given as negative in exponential form?

Find the fifth roots of $-3+3i$ in exponential form. My answers are: $$1.335e^{3i\pi/20}$$ $$1.335e^{11i\pi/20}$$ $$1.335e^{19i\pi/20}$$ $$1.335e^{27i\pi/20}$$ $$1.335e^{35i\pi/20}$$ Wolfram ...
1
vote
1answer
30 views

Multiplication using addition using logarithms

Multiplication by addition using logarithms is possible and took place in past using slide rule and log tables. Is it still used in software? Maybe sometimes it's faster to convert numbers and use ...
2
votes
0answers
35 views

Tweaking Reddit's Ranking Algorithm

This image explains how Reddit's Ranking algorithm works. As you know, Reddit is a very high traffic site. Therefore, the post rank decreases quite fast. This algorithm puts emphasis on bringing ...
8
votes
3answers
120 views

Logarithmic Integral II

While reviewing an old calculus book the following integral was assigned: \begin{align} \int_{0}^{1} \left( x^{a-1} - x^{n-a-1} \right) \, \frac{\ln^{2}x \, dx}{1-x^{n}} = \frac{2 \, \pi^{3} \, ...
4
votes
3answers
81 views

Simple Logarithms Equation

$$3^x = 3 - x$$ I have to prove that only one solution exists, and then find that one solution. My approach has been the following: $$\log 3^x = \log (3 - x)$$ $$x\log 3 = \log (3 - x)$$ $$\log 3 ...
1
vote
2answers
52 views

how to find log base 2 of decimal number without calculator

As with calculator things are simple but I don't know how to calculate log base 2 of decimal number without calculator. like $\log_2(0.25)$ etc.
1
vote
1answer
42 views

Requirement for a given function to be smooth

I have quite a basic question about the derivatives. My uncertainty comes mainly from the fact that I don't know how the complex logarithm behaves. Here is the description (this task is not ...
9
votes
1answer
159 views

For which complex $a,\,b,\,c$ does $(a^b)^c=a^{bc}$ hold?

Wolfram Mathematica simplifies $(a^b)^c$ to $a^{bc}$ only for positive real $a, b$ and $c$. See W|A output. I've previously been struggling to understand why does $\dfrac{\log(a^b)}{\log(a)}=b$ and ...
6
votes
5answers
81 views

Solve for $x$ : $\log_3(3x + 2) = \log_9(4x + 5)$

Solve for $x$ $$ \log_3(3x + 2) = \log_9(4x + 5) $$ I changed the bases of the logs $$ \frac {\log_{10}(3x + 2)} {\log_{10}(3)} = \frac {\log_{10}(4x + 5)} {\log_{10}(9)} $$ Now I'm stuck, ...
1
vote
0answers
34 views

Create log-normal on y axis?

I currently have a graph with log numbers on the x-axis and the y-axis goes from 0-100. How can I get it to I guess log normal y-axis as shown in the picture below? Thank you for your help!
1
vote
2answers
50 views

Show that $a^{\log_c b}= b^{\log_c a}$

Show that $a^{\log_c b}= b^{\log_c a}$. I start from LHS and add $\log a$ on it, but it leave $\log_c b$. Then I have not idea about how to continue it(maybe my working is wrong... Can anybody solve ...
2
votes
1answer
20 views

Are there more convenient ways of getting the number of digits of a positive integer?

I want to define $n$th power of $10$ for a positive integer. Say for $43$ it would be $2$, for $5$ it would be $1$, for $9999$ it would be $4$. As for $1$, $10$, $100$, ... I am still shifting between ...
6
votes
3answers
553 views

Basic Logarithm Equation

$\log_2(x) = \log_x(2) $ Using the change of base theorem: $\dfrac{\log(x)}{\log(2)} = \dfrac{\log(2)}{\log(x)}$ Multiplied the denominators on both sides: $\log(x)\log(x) = \log(2)\log(2)$ I kind ...
2
votes
3answers
78 views

a log inequality

Can anyone offer some guidance on proving the following inequality? Define $\Lambda_1(a)=-a\log a$ and $\Lambda_2(a,b)=-(a+b)\log(a+b)$. Then if $a$, $b$, $c$, and $d$ are non-negative numbers summing ...
1
vote
7answers
160 views

How do I solve the equation $e^{\ln(2x+1)} = 5x$?

The problem is $$e^{\ln(2x+1)} =5x$$ I've tried using natural logs to both sides like.. $2x+1= \ln 5x $ But I'm not sure if $\ln$ and $e^{\ln}$ cancel out.
0
votes
4answers
59 views

Given $x^2 + y^2 = 34xy$, show that $\log\left(\frac{x+y}{6}\right)= \frac{\log x + \log y}{2}$

If $x^2 + y^2 = 34xy$, show that $$\log\left(\frac{x+y}6\right)= \frac{\log x + \log y}{2}.$$ I tried to put log into the first equation, but I have no idea about how the $34$ being simplified in the ...
0
votes
2answers
34 views

If $3(4^h)=4(2^k)$ and $9(8^h)=20(4^k)$,show that $2^h = \frac{4}{5}$

If $3(4^h)=4(2^k)$ and $9(8^h)=20(4^k)$,show that $2^h = \frac{4}{5}$. I tried to substitute the equation 1 into equation 2 so that I can find the value of $k$ or $h$, but it did not work as the base ...
1
vote
1answer
26 views

Simple Logarithm and JavaScript Question

I have a simple formula that I am trying to convert to JavaScript, I'm just stuck trying to reverse it. My math skills have deteriorated over the last few years and im stuck. Here is the formula ...
-2
votes
1answer
25 views

Natural log problem divide by zero problem for stock/fx contributions

The Stock Price move from 100 ($p_1$) to 150 ($p_2$) and the FX rate moves from 1.2 ($c_1$) to 0.8 ($c_2$). therefore the base currency value stays the same. I am looking for the fx vs stock ...
0
votes
3answers
83 views

How to find $\log{x}$ close to exact value in two digits with these methods?

I'm trying to find the result of $\log{x}$ (base 10) close to exact value in two digits with these methods: The methods below are doing by hand. I appreciate you all who already give answers for ...
4
votes
3answers
133 views

How to solve a system of logarithmic equations?

I need to create a function with the following properties: $$f(1)=1$$ $$f(65)=75$$ $$f(100)=100$$ Additionally, the function needs to grow logarithmically. So that gives three equations: $$A \cdot ...
2
votes
2answers
64 views

Number of solutions of $a^{3}+2^{a+1}=a^4$.

Find the number of solutions of the following equation $$a^{3}+2^{a+1}=a^4,\ \ 1\leq a\leq 99,\ \ a\in\mathbb{N}$$. I tried , $$a^{3}+2^{a+1}=a^4\\ 2^{a+1}=a^4-a^{3}\\ 2^{a+1}=a^{3}(a-1)\\ ...
1
vote
3answers
39 views

Logarithmic equation with logarithm in power.

$$x^{\log_{\,3}(3x)}=9$$ I tried to turn the exponential to logarithm form $- \log_{\,x}(9) = \log_{\,x}(3x)$. I also tried using the property $a=\log_{\,b}(b^a)$, but it didn't get me anywhere. I ...
6
votes
3answers
99 views

Derivative Of $\ln(x)$

It is required to find the derivative of the natural logarithm of $x$: $\frac {d}{dx}\ln(x)$ My solution: Let $f(x)=\ln(x) $ then $f'(x)=\frac {d}{dx}\ln(x) $ By definition:$$f'(x)= \lim_{h\to ...
0
votes
4answers
45 views

Find value of $x$ in a logarithmic equation

If $$2^{(\log_{2}3)^x} = 3^{( \log_3 2)^x}$$ then what is the value of $x$ in this equation? Could taking log on both sides help?
2
votes
1answer
69 views

Solving $x - a \log(x)=b$

Let $a>0$ and $b \in \mathbb{R}$: Assume there exists an $x >0 $ s.t. $$x - a\log(x) = b$$ holds. How can it be determined in closed-form?
1
vote
2answers
51 views

How to calculate $\log(x) = 1/2\log(16) - 1/3\log(8) + 1$

This is my first question. This is basic math, but what I get does not match the alternatives I have. So I was wondering if I did something wrong. Step 1: $\log(x) = 1/2\log(16) - 1/3\log(8) + 1$ ...
1
vote
2answers
72 views

Solve this exponential equation: $3^{2x}+\left(\frac{1}{2}\right)^{-x} \cdot 3^{x+1}-2^{2x+1}=0$

I tried solving this equation $$3^{2x}+\left(\frac{1}{2}\right)^{-x} \cdot 3^{x+1}-2^{2x+2}=0$$ by taking the log of both sides, but with no results, what do I do? Sorry if this equation is very easy, ...
2
votes
3answers
73 views

How to solve $\log(x -1) + \log(x - 2) = 2?$

I'm doing this exercise: $$\log(x - 1) + \log(x - 2) = 2$$ My steps: Step 1: $$\log(x-1)(x - 2) = 2$$ Step 2: $$(x - 1)(x - 2) = 10^2$$ Step 3: $$x^2 - 3x + 2 = 100$$ Step 4: $$x^2 = 98 + ...
4
votes
4answers
147 views

Trouble evaluating the sum involving logarithm

I was trying to solve this problem: Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$ In the procedure I followed, I came across the following sum: ...
0
votes
2answers
47 views

Splitting the summation sign

i am trying to understand the second step in the formula per below - and how the summation sign $\sum_{k=1}^K$ splits into the terms 1-$\sum_{k=1}^{K-1}$ terms. Any help much appreciated
2
votes
2answers
396 views

How to compare logs without a calculator

I have multiple different log sums that I need to evaluate. How would I calculate the following without using a calculator or log tables?
0
votes
2answers
52 views

Integral of logarithm of exponential function

I am trying to solve this integral: $$\int \log\left(1 + \frac{1}{\pi}\exp\left(\frac{-x^2}{2a^2}\right)\right) dx$$ where $a$ is some fixed constant. The bounds of this integral are $-a$ and $a$, ...
1
vote
1answer
48 views

Which “approximate” value of f(0.98) is this question looking for?

In a section of a calculus workbook dealing with local linearity and linear approximations of functions, the following question is posed: Consider the function f(x) = aln(x+2). Given that f'(1) = ...
0
votes
1answer
34 views

Solving logarthmic inequality

Question Find integral solutions of this inequality$$\left (\frac{1}{10}\right )^{\log_{x-3}^{x^2-4x+3}} \ge 1$$ My try : I took log on both sides and got $\log_{x-3}^{x-1} \le-1$ but ...
4
votes
1answer
4k views

How to prove if log is rational/irrational

I'm an English major, now doubling in computer science. The first course I'm taking is Discrete Mathematics for Computer Science, using the MIT 6.042 textbook. Within the first chapter of the book's ...
1
vote
3answers
55 views

Logarithm problem with two bases

Given $$\log_x9+\log_9x=\dfrac{10}3.$$ How can I find the greatest value of $x$ that satisfies the equation above?
-1
votes
2answers
24 views

Equation solving involving logaritm

i need help solving this equation to find the variable RC: Vc=Vin(1-e^-t/RC) I already know Vc, Vin and t. I always get it wrong so the RC is negative. It represent time so it shouldn't be. Thanks! ...
2
votes
1answer
32 views

How do I find a point on a graph which is equal on both the axis?

I have the equation $ 10^{x-0.7711} = x $. In order to find x, I thought that I'll graph the equation, and the point where x = y, will be the answer. How do I do this? Or is there any other way to ...
1
vote
0answers
56 views

Integrals of the type $f'(z)/f(z)$

I am having trouble understanding integrals of the form: $$\int_\gamma\frac{f'(z)}{f(z)}\,{\rm d}z$$I am aware that there are problems with the complex logarithm, and we have the formula: ...
1
vote
1answer
65 views

Describe the Riemann surface:

$$W = \sqrt{1-z^2}$$ I would like hints only. Using @Dr.MV's hint, I get two factors: the first is $$\sqrt{(x-1)+y^2}^{\frac{1}{2}}e^{i\frac{\theta}{2}}$$, which, when we let theta range from 0 to ...
2
votes
2answers
69 views

HowTo solve this integral involving logarithm

I would like to solve integrals of the form $$I(c) := \int_0^\infty \log(1+x) x^{-c} \, dx ,$$ where $c \in (1,2)$. Mathematica says either 1) $I(c) = \frac{\pi}{1-c} \csc(\pi c)$ or 2) $I(c) = ...
1
vote
2answers
93 views

$\ln{\left(\frac{1}{0}\right)} = -\infty$?

I have shown it using a theorem that I made, but I am not sure, as $\lim_{\alpha \to 0^{-}}{\left(\frac{1}{\alpha}\right)} = -\infty$, and $\lim_{\alpha \to 0^{+}}{\left(\frac{1}{\alpha}\right)} = ...
-1
votes
2answers
69 views

Is the following equality true? : $\log\left(-\left(\frac{x+1}{x-1}\right)^3\right)=3\log\left(\frac{1+x}{1-x}\right)$

Is the equality below true over all complex numbers? $$\log\left(\frac{1+\frac{3x+x^3}{1+3x^2}}{1-\frac{3x+x^3}{1+3x^2}}\right)=3\log\left(\frac{1+x}{1-x}\right)$$ The L.H.S. (Left hand side ...