Questions related to real and complex logarithms.

learn more… | top users | synonyms

-1
votes
2answers
29 views

What is the Solution? x ≥8lgx

x ≥ 8lgx I have to find which x satisfy this inequality. I found the points using graph, but I'd like someone to show me how to find it without it.
3
votes
1answer
139 views

Find the number $n^{2}$ from the number $n^{n^{n^{2}}}$

Find the number $n^{2}$ from the number $\large n^{n^{n^{2}}}$ Any help? I tried with $\log$ but I got nothing.
1
vote
2answers
44 views

Show $n \cdot \log_b r + \log_b \frac{r}{r-1} \le \lceil n \cdot \log_b r \rceil$

Let $n$ be a natural number and $b, r > 1$ be two natural numbers, then I guess we have $$ n \cdot \log_b r + \log_b \frac{r}{r-1} \le \lceil n \cdot \log_b r \rceil. $$ where $\lceil x \rceil = ...
1
vote
1answer
42 views

Logarithm Propr

I'm having a bit of trouble proving the following property: Theorem If $Re(z)>0 $ and $ Re(w)≥0$, then $\log(zw)=\log(z)+\log(w)$, where log is the principal branch. I know that $\log (zw) = ...
3
votes
3answers
199 views

Indefinite integration: $\int x^{x^2+1}(2\ln x+1)dx$

Find the value of the integral: $$\int x^{x^2+1}(2\ln x+1)dx.$$ My attempt: I tried by using integration by parts, but not working since $x^{x^2+1}$ keeps coming again and again. Then I tried putting ...
0
votes
1answer
27 views

Linear, Squared and Logarithmic scales with given input domain and output range

The input domain is $[12,24]$ and the output range is $[0,720]$. Now I know that with using linear scaling the value $16$ of the input range is mapped to $240$; with using sqrt scaling the same value ...
7
votes
4answers
703 views

Limit to infinity with natural logarithms

I found the following problem in my calculus book: Solve: $$\lim_{x\to \infty } \left(\frac{\ln (2 x)}{\ln (x)}\right)^{\ln (x)} $$ I tried to solve it using log rules and l'Hôpital's rule with no ...
0
votes
1answer
22 views

Minimum of the function $b\log_b x$

Why the function $b\log_b x$ has its minimum at $b=e$? How to explain this? I'm asking because I can't understand why ternary base has more economy than binary: ...
0
votes
2answers
38 views

Logarithms and Taylor Series

Before Log Tables, how were they able to compute expressions such as $2^{2.221}$? I understand they could take a Taylor expansion of $\frac{1}{x}$, but how were they able to condense the expansion ...
3
votes
4answers
80 views

Solve for $x$ - Logarithm Equation $\ln x+\ln(x+1)=\ln 2$

My attempt: $\ln x(x+1)=\ln 2$ $e^{\ln x(x+1)}=e^{\ln 2}$ $x(x+1)=2$ $x^2+x-2=0$ $(x-1)(x+2)=0$ therefore $x=1, -2$
0
votes
1answer
17 views

Geometric distribution with given probability value.

The probability of a man hitting a target is $2/3$. If he doesn't stop shooting until he hits the target for the first time, a) What is the probability of taking 5 shots to hit the target? b) Which is ...
-3
votes
1answer
36 views

Express as a single logarithm [closed]

Hi I need to express the following and have no clue how to do so. $$\ln(x+3)-3\ln(x-7)-\ln(x+8)$$ Can someone please help
1
vote
5answers
80 views

logarithm proof fallacious or not?

$e^{-x}=e^{1/x}$ Taking the natural logarithm of both sides $$\ln(e^{-x})=\ln(e^{1/x})$$ $$-x=1/x$$ $$-x^2=1$$ $$x^2=-1$$ $$x=i$$ I know I am doing something wrong here. Also can someone please ...
5
votes
3answers
41 views

Logs - Simplifying with arbitrary constant

I've tried simplifying my answer, which has a constant in it. I would like to know if I am on the right track: $$ \ln(y) = -{x^2\over 2y^2} + C $$ C can be considered as an Arbitrary Constant. From ...
2
votes
5answers
110 views

how to prove that $\ln(1+x)< x$

I want to prove that: $\ln(x+1)< x$. My idea is to define: $f(x) = \ln(x+1) - x$, so: $f'(x) = \dfrac1{1+x} - 1 = \dfrac{-x}{1+x} < 0, \text{ for }x >0$. Which leads to $f(x)<f(0)$, ...
25
votes
2answers
546 views

A difficult logarithmic integral ${\Large\int}_0^1\log(x)\,\log(2+x)\,\log(1+x)\,\log\left(1+x^{-1}\right)dx$

A friend of mine shared this problem with me. As he was told, this integral can be evaluated in a closed form (the result may involve polylogarithms). Despite all our efforts, so far we have not ...
1
vote
1answer
31 views

Imprecise logarithms that reference sets of numbers.

I apologize in advance if my question seems vague, I'm only in algebra II, so It may turn out that I lack the terminology to phrase my question correctly. Some background, we just finished our unit ...
1
vote
0answers
60 views

Are logarithms of prime numbers algebraically independent?

From Baker's theorem it follows that a linear combination of natural logarithms of prime numbers with non-zero algebraic coefficients can never be zero. Has it been proved that the set of all natural ...
1
vote
2answers
51 views

Summation of a logarithmic series for $\ln(2(r^2 - 1)/r^2)$

Given that $$\sum_{r=2}^{n}\ln\frac{r^2-1}{r^2}=\ln\frac{n+1}{2n}$$ for $n >1$. Express $$\sum_{r=32}^{62}{\ln\frac{2(r^2-1)}{r^2}}$$ as $$A\ln 2 + B\ln3 + C\ln7$$ where $A$, $B$, $C$ are positive ...
0
votes
1answer
36 views

How to solve integral with natural logarithm and product

I am trying to solve the following integral: $$\int{\frac{x}{4} \ln\left(\frac{4}{x}\right)}$$ Using this integral table, the more close case is (43). However, this is not the right one to use. Do ...
9
votes
2answers
187 views

Checking logarithm inequality.

Which one of the following is true. $(a.)\ \log_{17} 298=\log_{19} 375 \quad \quad \quad \quad (b.)\ \log_{17} 298<\log_{19} 375\\ (c.)\ \log_{17} 298>\log_{19} 375 \quad \quad ...
2
votes
1answer
38 views

Why does this sequence of operations give $x^{\frac{1}{x-1}}$?

I found (purely from experimentation) that if you start with a random number and successively: Exponentiate, Raise to the power of $x$, Take the log with the same base as step one, Take the $x$-th ...
11
votes
0answers
120 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\\ ...
4
votes
2answers
52 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 ...
12
votes
2answers
202 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 ...
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 ...
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 ...
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 ...
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 ...
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} \, ...
1
vote
2answers
51 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 ...
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
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!
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
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
551 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 ...
1
vote
7answers
155 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 ...
2
votes
3answers
77 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
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 ...
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
71 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, ...