Questions related to real and complex logarithms.

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5
votes
3answers
494 views

How much proof is needed in such paper (Maths related)?

I'm writing a paper (report) regarding Euler's Number $\space e \space$ (even though he didn't discover it). Within this paper, I show that: $${d\over dx} {e^x} = {e^x}$$ **NOTE: ** This is not ...
0
votes
1answer
87 views

Logarithm in the exponent

$$(2x)^{\log 2} = (3y)^{\log 3} \\ 3^{\log x} = 2^{\log y}$$ Solve for $x$ and $y$. My intuition for solving such problems is taking the logarithm on both sides but it does not work. I also ...
0
votes
1answer
28 views

What is the value of $x$ in this logarithmic inequality? [closed]

Please help me with this inequality : $$\log_2 (x^2-2x) - 3 >0 $$
0
votes
2answers
70 views

If the integral of $c/x$ is $c.log(x)+C$ what is the base?

This question is a follow up to an answer I gave here: How to integrate $1/x$? After the algebra I said that 'This step of course gives the argument of $ln()$ the value $e$ and note that so far we ...
0
votes
1answer
16 views

Calculating the amount of times a binary search could run (worse case) without a calculator/calculating base 2 logs without a calculator.

Ok so I had a question on a test that I had to do without a calculator. And I can not figure out how in the world I am supposed to do it without a calculator. The question asked to find how many ...
1
vote
2answers
41 views

Logarithm question with base change

If $\log_{12} 27 = a$ then find the value of $\log_6 16$.
-2
votes
3answers
58 views

Can someone please explain how $60+\ln(64)-\ln(8)$ is equal to $60+\ln(8)$ [closed]

Can someone please explain how $60+ \ln(64)- \ln(8)$ is equal to $60+\ln(8)$. I can't understand why this is true.
0
votes
2answers
44 views

what is the value of $x$ in this logarithmic question? [closed]

What is the value of $x$: \begin{equation} x^{\log_5 x} >5 \end{equation} Thanks for the help.
0
votes
3answers
53 views

Is $n^\frac{1}{10} \in O((\log n)^{10})$?

This question came up in a recent discussion: is $n^\frac{1}{10} \in O((\log n)^{10})$? First time I've come across a power of a log in a long time, and as far as I recall, there are no identities ...
0
votes
0answers
37 views

A fast converging limit for $\ln x$ (or why $\ln 2 \approx \sqrt{\sqrt{42}-6})$

I've read a lot about approximating logarithms recently, and apparently it's not easy. It can be done by Taylor series (slow convergence), by continued fractions (also slow) and also by some limits. ...
0
votes
2answers
63 views

I need to integrate $\ln(x^2)$ but I can't seem to get it right.

I have an issue that requires me to integrate $\ln(x^2)$ and I know it's done through integration by parts, but I just can't seem to get it right. Does anyone know how it would work out?
0
votes
0answers
13 views

Generator Powers

I have factor base such that $$B=\{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47\}$$ I need to construct a program which inputs a prime "p" (p>72) and an integer "a", then outputs the 15 distinct values ...
1
vote
0answers
36 views

Newtons Law Of Cooling (And Heating)

Rule is: $D= A.e^{-kt}$, Where: $k,a$ are elements of real numbers, $D$ is the difference between the temperature of the item and the surrounding air, and t is the time in hours since the object ...
0
votes
1answer
26 views

$\log_{10}y = m\log_{10} x + \log_{10} c $ for straight line

Express $x$ in terms of $y$: $\log_{10}y = 2\log_{10}x + \log_{10} c$ When $x = 0$ $$2 = \log_{10} c$$ $$c = 100$$ $$\log_{10} y = 2\log_{10}x + \log_{10} c$$ $$\log_{10}y = \log_{10}(x^2c)$$ ...
1
vote
1answer
31 views

Solve $\log_9(x-4) - \log_9(x-8)= \frac{1}2$

Solve $\log_9(x-4) - \log_9(x-8)= \frac{1}2$ $(x-4) - (x-8)= 9^\frac{1}2$ $(x-4) - (x-8)= 3$ The answer is 10 but I am not sure how that was obtained.
0
votes
1answer
58 views

How to solve $ax + b \log x =c$ or $\frac{a}{x}+b\log x=c$?

Here $a,b,c$ are any real numbers. We can use graphical methods using Mathematical tools, but what are the other techniques ?
0
votes
1answer
26 views

write $\log_42x$ in the form $y =\log_4x+k$

write $\log_42x$ in the form $y = \log_4x+k$ I take it this is one of the log rules but I don't see which and I do not understand where k comes from or what the constant stands for.
-1
votes
1answer
48 views

Find the value of 'x'

What is the value of x, if: $$(\log_{10} 4 )^2 + (\log_{10} 4 )^4+ (\log_{10} 4 )^{16}+ (\log_{10} 4 )^x = 6? $$
0
votes
0answers
16 views

Estimating elasticity of y with respect to x in a log-log specification

The question My rudimentary workings so far is that; log(y_i/x_i) = log(y_i)-log(x_i) Factorise, so, log(y_i/x_i) = log(y_i) + upsilon_i - log(gamma_i + 1) Thus, elasticity of y to x is always >1 ...
2
votes
2answers
176 views

How can I rearrange this logarithmic formula to be computer friendly?

I've had a look through the logarithmic identities on Wikipedia, but nothing fits the bill. Basically, I have a formula which shows how much more 'risky' one number is compared to another, where 0 = ...
3
votes
3answers
76 views

Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$

I was given a task to prove that inequality is true for x>0: $(e^x-1)\ln(1+x) > x^2$. I've tried to use derivatives to show that the $f(x) = (e^x-1)\ln(1+x)-x^2$ is greater than zero, but has never ...
0
votes
1answer
40 views

How to simplify this ln?

I am solving this problem $$\int {\frac{1}{\sqrt{9x^{2}-25}}dx}$$ the step I got stuck, $$\frac13 \ln{\left(\frac{3x+\sqrt{9x^2-25}}{3}\right)} $$ and in my textbook the answer was $\frac13 ...
15
votes
3answers
2k views

Why is Euler's number used as a base for logarithms? [duplicate]

Is there some special property of '$e$' which makes it suitable to be used as a base for logarithms? Moreover, does the natural logarithm possess some advantage over the common logarithm? I don't ...
2
votes
1answer
17 views

How to find the highest [natural] radix base of a given number with a natural output

Like the title says, I'm trying to make a program that finds the highest natural radix of a given number with a natural output. My program works, but it loops every number possible number up to a ...
2
votes
2answers
32 views

Prove $\log u > \frac{u - 1}{u}$ for $u > 1$

How to prove that for $u > 1$ $$\log u > \frac{u - 1}{u}$$ without using integrals? I think I'm supposed to use derivatives or Taylor's theorem, as the exercise comes from a lecture about these ...
1
vote
1answer
24 views

Omitting the refference to a particular logarithmic base - order notation

How can I prove by using the Order notation definition that we can conventionally refer to an algorithm taking "log time", without referring to a particular logarithmic base?
0
votes
1answer
71 views

How is this logarithm problem solved?

If $\dfrac{xy\log xy}{x+y} = \dfrac{yz\log yz}{y+z} = \dfrac{zx\log zx}{z+x}$ show that $x^x = y^y = z^z$. I tried equating to a constant $k$ and adding up I also tried adding up the numerators and ...
2
votes
1answer
37 views

Find the possible values of $x$ if $2^{2x+1} = 3(2^x) -1$

Find the possible values of $x$ if $2^{2x+1} = 3(2^x) -1$ I know that $x=0$ and $x=-1$ are possible values of $x$ by looking at the equation. I need help understanding how to use logarithms to ...
0
votes
1answer
48 views

Differentiating a Log Function [closed]

My current $\ln$ function is: $$ \ln\Big(\frac{y}{x}\Big) = b_1 + b_2 ln(x). $$ How would I find the elasticity for this model? What I have done so far is convert the equation to: $$ ...
2
votes
1answer
46 views

What's wrong with this wrong derivation of $\ln(-x)$?

If we take $\ln(x) = \int_{1}^{x} \frac{1}{y} dy$ as the definition of logarithm, then I am ending with a stupid derivation of $\ln(-x)= - \int_{-x}^{-1} \frac{1}{y} dy$. From the definition, if I go ...
3
votes
1answer
36 views

Density of $\{ \ \{\ln k\} \ \}_{k=1}^{\infty}$

Is this sequence dense in $(0,1)$? I want to say that it is, I think the transcendence of the logarithm function is leading me to believe that it is, but I don't know how to prove it.
0
votes
0answers
16 views

logarithm of odds score at test value of 0

In my human genetics class, we learned to compute a logarithm of the odds (LOD) score as follows: $LOD=\log_{10}{\frac{\theta^r(1-\theta)^{n-r}}{0.5^n} }$, where $\theta$ is a test value for ...
1
vote
2answers
34 views

Calculating the discrete logarithm

I'm given a prime number $p = 1217$ I'm also given the following equations: $ 40 = \log2 \mod 64 $ $ 63 = \log3 \mod 64 $ $ 13 = \log5 \mod 64 $ $ 13 = \log2 \mod 19 $ $ 10 = \log3 \mod 19 $ $ ...
0
votes
2answers
26 views

How do I solve for x? Do I need the Lambert W function?

I need to solve the next equation x: $d-x+yln[\frac{d}{x}]=b$ y, d, b, and x are all real, positive numbers. How do I solve for x? Do use the lambert W function and if so how is that done? Thanks! ...
0
votes
2answers
29 views

Rewrite a Logarithm as a product of logarithms

Can anyone help me to understand this? $$\log_2 n = \log_2e \log n$$
-2
votes
1answer
43 views

Difficulty finding point of inflection

The Problem is... $N(t)=\frac{200,000}{1+999e^{-0.4t}}$ Use logarithmic differentiation to find the time to the point of inflection. I know that in order to find the point of inflection I must set ...
0
votes
0answers
16 views

Find the complex potential given the real part $\frac{6000}{\pi}\theta$

In my 'Complex Variable Theory' notes, we have an example where we use a linear fractional mapping to solve for Laplace's equation between semi-circular plates. The Question reads: Find the ...
1
vote
1answer
33 views

Limit with logarithm: $\lim_{n \to \infty} \frac{n^\alpha}{\ln^\beta n}$

What is the limit $\lim_{n \to \infty} \frac{n^\alpha}{\ln^\beta n }$ (ln=natural logarithm) for alfa real and less than zero? I found out it is zero for $\beta\ge0$, since then you can use the ...
0
votes
1answer
40 views

Please explain the logarithmic equation

The default equation is $(1 + x)^3=4^{-y^2} $ I solved as follows: $(-3\log_4(1+x))^{1/2}=y$ With the logarithm base equal to $4$, my idea is that $4$ is the number we have in the right part of ...
1
vote
2answers
71 views

$2\ln(-x) \neq \ln(-x^2) ? $

I know the rule $$n\ln(x) = \ln(x^n)$$ But this doesn't apply to $$2\ln(-x) = \ln(-x^2)$$ Can you see what I'm not understanding?
1
vote
0answers
12 views

Financial Mathematics--Finding Compounding Period given Annual and Effective Interest Rates

I'm trying to find a compounding period C when given an annual interest rate r and effective annual yield i. I'm working with the following equation: $i=(1+r/C)^C-1$ I'm having trouble re-writing ...
1
vote
4answers
49 views

How to prove that $(x+c)\log(\frac{c+x}{x})>c$

How to prove that $(x+c)\log(\frac{c+x}{x})>c$ for $x, c > 0$? For $\frac{c+x}{x} \ge e$ it's obvious.
1
vote
3answers
35 views

Limit of $f(x)=|\log x|$

My textbook solved this problem: Find $f'(1^{-})$ if $$f(x)=|\log x|$$ for the interval $x>0$ The textbook solved it by using the method described below: $$f'(1^{-})=\lim\limits_{x\to 1^{-}} ...
0
votes
0answers
26 views

Simplify/expand $\ln \left(\sum^n_{i=1}x_i^{\theta-1}\right)$

Can someone help simplify/expand this natural log? I want to bring the $\theta - 1$ down in front of the $\ln$ but I don't know how the rules work with the summation. $$\ln\left( ...
1
vote
0answers
74 views

Hard Logarithm Integral [duplicate]

I recently had this integral in my homework and don't really know how to proceed. $\int^1_0\frac{\ln (1+x^{2+ \sqrt3})}{x+1}dx$ So far I figured that $ \int^1_0\frac{\ln (1+x^{a})}{x+1}dx= ...
4
votes
1answer
95 views

How to compute $\int 1/x \, dx$ without knowing its anti-derivative

How do you compute $$\int\frac 1x \, dx$$ without knowing its anti-derivative to start with? Is there a way to do it by parts or substitution?
1
vote
0answers
36 views

Really simple question: Add $\bar4.74628$ and $ 3.42367$ .I just need to cross check answer.

Add $\bar4.74628$ and $ 3.42367$ This question is about characteristics and Mantissa.I thought my book has written the wrong answer in the example.I just wish to cross check because this seem like ...
2
votes
2answers
28 views

Natural logarithm power notation

I am trying to understand how to use Dirichlet's test for convergence and saw an example here (example 2). Show that $\displaystyle\sum_{i=1}^\infty \frac{2^{2n}n^2}{e^n\,n!}\frac{1}{\ln^2n}$ ...
0
votes
1answer
30 views

Adding logarithms with different bases

Just had an exam, this sinister question I know I did wrong lingers in my mind: Solve for $x$, $$2-\log_3(x-7) = \log_{\frac{1}{3}} (2x)$$ On phone not sure how to write the equation properly. ...
-1
votes
1answer
22 views

Big O notaion O(n) and logaritms [closed]

Can someone explain me the subjects Big O notation and logarithms please? I can't understand those subjects For example if I have a question like this: recall that logan is the power to which you ...