Questions related to real and complex logarithms.

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1answer
41 views

Natural Logs and Anit-Derivatives are kicking me

I am given a problem involving rates of flow, $F(t)=\frac{t+7}{2+t}$ is the rate at which a bucket is being filled. The same bucket is being emptied at a rate given by $E(t)=\frac{\ln(t+4)}{t+2}$. My ...
3
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3answers
77 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 ...
3
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3answers
39 views

Trying to show that $\ln(x) = \lim_{n\to\infty} n(x^{1/n} -1)$

How do I show that $\ln(x) = \lim_{n\to\infty} n (x^{1/n} - 1)$? I ran into this identity on this stackoverflow question. I haven't been able to find any proof online and my efforts to get from ...
1
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1answer
142 views

Forward Algorithm Hidden Markov Model matrix help [Discrete]!

So this may seem like a bioinformatics question but it is the math part that is giving me trouble. I'm using a Python package called YAHMM to model DNA sequences. I created a model with two states ...
0
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1answer
12 views

Formula for calculating markup with big % for small amounts and small % for larger amounts

I am trying to come up with a formula for calculating markup for products that range in value from a few cents up to tens of Dollars. At 10c I would like the markup to be around 500%, and from 2 ...
4
votes
3answers
148 views

Prove that $f(ab) = f(a) + f(b)$

Question : Assume only that $f: (0,\infty)\to{\mathbb{R}}$ is differentiable and that $f'(x) = 1/x$, and $f(1)=0$. Prove that for all $a,b \in(0,\infty)$, $f(ab)=f(a)+f(b)$. [Hint: Let $g(x)=f(ax)$] ...
5
votes
3answers
255 views

Proving $x$ is a given quotient of logarithms

I'm practicing some questions on logarithms at the moment in order that I'm up to speed with the problem solving aspect before I embark on my PHD in chemical engineering at Boston college next year. ...
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5answers
49 views

Multiplying two logarithms (Solved)

I was wondering how one would multiply two logarithms together? Say, for example, that I had: $$\log x·\log 2x < 0$$ How would one solve this? And if it weren't possible, what would its domain ...
0
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1answer
12 views

Exhaustive search times: 2 to power k = 100 hours - double k, how many hours

An exhaustive search (i.e. checking all combinations of values) takes 100 hours to go through all permutations where a binary key has a length of k. $2^k$ = 100 hours where k is the number of digits ...
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0answers
43 views

log(x,2log(2x, 2log(2,4x))) >1 , find answers of x

how i slove it , please help me? log(x,2log(2x, 2log(2,4x))) >1 my try: if x>1 =>2log(2x, 2log(2,4x))>x => 2log(2,4x)>(2x)^(x/2) =>4x>2^(((2x)^(x/2))/2) another way log(x,2log(2x, ...
11
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1answer
70 views

If $\log_35=a$ and $\log_54=b$, what is $\log_{60}70$?

One student sent me this question: If $\log_35=a$ and $\log_54=b$, what is $\log_{60}70$? Question asks the value of $\log_{60}70$ in terms of $a$ and $b$. Equations for $a$ and $b$ involved ...
10
votes
5answers
142 views

How do you solve $x^2 = \left(\frac 12\right)^x $?

I'm having trouble finding the steps to solve for $x$. The solutions to this equation are $x=-4$, $x=-2$, and $x=0.76666$ when solved graphically and through the solve function of a TI-nspire cx CAS. ...
10
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3answers
463 views

Number system with $e^x = 0$ for some $x$

It is well known that $e^x \ne 0$ for all $x \in \mathbb{R}$ as well as $x \in \mathbb{C}$. Upon reading this article and doing a bit of research I have found that this also applies to the ...
1
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2answers
24 views

Cancelling a logarithm

I was wondering if there was a way to cancel out a logarithm? For example: $\log_a A$ > $\log_a B$ What would a have to be for the log to go away and be left with A > B? Thanks in advance!
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2answers
43 views

So many logs with different bases

$ \large { 6 }^{ \log _{ 5 }{ x } }\log _{ 3 }( { x }^{ 5 } ) -{ 5 }^{ \log _{ 6 }{ 6x } }\log _{ 3 }{ \frac { x }{ 3 } } ={ 6 }^{ \log _{ 5 }{ 5x } }-{ 5 }^{ \log _{ 6 }{ x } }$ The sum of ...
0
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1answer
19 views

Find unknown x coordinate from log graph

I am not sure where to start on this question. I am not sure how to fit the coordinates into the equation $y=\log_3(x-4)$
1
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1answer
40 views

How to prove derivative of logarithm with base $b$?

I learned how to derive a logarithm with any base. This is the formula: $$\frac{d}{dx}\log_bx=\frac{1}{x\ln b}$$ How can it be proved?
2
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1answer
94 views

How $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots =\ln 2$? [duplicate]

while doing the Integration problem using Limit of a sum approach i have a doubt how $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots =\ln2$$ by infinite geometric series we have ...
0
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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 ...
5
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3answers
495 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 ...
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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 ...
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1answer
28 views

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

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

Logarithm question with base change

If $\log_{12} 27 = a$ then find the value of $\log_6 16$.
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6answers
9k views

Calculate logarithms by hand

I'm thinking of making a table of logarithms ranging from 100-999 with 5 significant digits. By pen and paper that is. I'm doing this old school. What first came to mind was to use $\log(ab) = ...
0
votes
1answer
17 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 ...
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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.
1
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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.
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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
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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
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0answers
38 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
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2answers
64 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?
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1answer
48 views

Find the value of 'x'

What is the value of x, if: $$(\log_{10} 4 )^2 + (\log_{10} 4 )^4+ (\log_{10} 4 )^{16}+ (\log_{10} 4 )^x = 6? $$
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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 ...
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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
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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 $x = 0$ $$2 = \log_{10} c$$ $$c = 100$$ $$\log_{10} y = 2\log_{10}x + \log_{10} c$$ $$\log_{10}y = \log_{10}(x^2c)$$ ...
0
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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
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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.
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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
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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 = ...
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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 ...
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 ...
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2answers
48 views

Lambert W function with natural log

I need to solve the next equation x: $d-x+yln[\frac{d}{x}]=b$ I inserted this into Wolfram Alpha and it returned: $x = y \Bbb{W}[\frac{e^\frac{d-b}{y}d}{y})]$ y, d, b, and x are all real, ...
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 ...
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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?
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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 ...
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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 ...
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.
2
votes
1answer
38 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 ...