Questions related to real and complex logarithms.

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2
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3answers
42 views

Prove that for any positive integer $n$ and $d$, $\sum_{k=0}^d 2^k\log_2(\frac{n}{2^k})=2^{d+1}\log_2(\frac{n}{2^{d-1}})-2-\log_2{n}$

I could prove it by induction, but I need to see how I might have discovered it for myself (cause that's what's gonna be on exam).
0
votes
2answers
39 views

All real number solutions of equation $\log_{2011}(2010x) = \log_{2010}(2011x)$ are in certain interval. Which one is it?

This task has to be done with no calculator, but I don't have basic idea how to start. Can someone give me advice, I know this is pretty easy but I need direction for particularly this one? EDIT: I do ...
2
votes
2answers
45 views

Value of $\frac{1}{\log_aabc}+\frac{1}{\log_babc}+\frac{1}{\log_cabc}$

How to find the value of $\frac{1}{\log_aabc}+\frac{1}{\log_babc}+\frac{1}{\log_cabc}$? I guess the answer will be $1$. But I don't know how to evaluate it. Can someone give me some tips?
1
vote
2answers
25 views

Assuming $d+1 <= log_2(n)$, how to show $d - 1 > log_2(n/8)$?

Also we know $d = log_2(n/2)$ rounded down to its nearest integer. Add (-2) to each side $$d-1 <= log_2(n) - 2$$ $$d-1 <= log_2(n) - log_2(4)$$ $$d-1 <= log_2(n/4)$$ This is as far as I can ...
4
votes
3answers
168 views

$a=b^x+c^x$, How to solve for $x$?

If $a=b^x$, then $x$ could be written in terms of $a$ and $b$; $x=\dfrac{\log(a)}{\log(b)}$. What about $a=b^x+c^x$? Could $x$ be written in terms of $a, b$ and $c$? $x={}$?
3
votes
4answers
143 views

How to compute the derivative of $\sqrt{x}^{\sqrt{x}}$?

I know have the final answer and know I need to use the natural log but I'm confused about why that is. Could someone walk through it step by step?
2
votes
2answers
38 views

Simplifying the equation $\log y = 10 + 0.5x$

Solve for $y$. When expressed in simplest form, what familiar kind of equation results? $$\log y = 10 + 0.5x$$ For this question, I would get rid of log first right? So, I would get ...
0
votes
2answers
24 views

Comparing the greatest values of two functions (Derivatives)

I've tried doing this task, and for this kind of task I should be using derivatives. When I done all the calculus, everything I got were some weird result which I do not know how to compare. Task ...
0
votes
1answer
76 views

Exponential & logarithmic functions [closed]

How can you solve the following logarithmic exponential expression for $y$ in terms of $x$, Where both $x$ and $y$ are positive distinct real numbers and $0<x<y <1$ ...
0
votes
1answer
50 views

What's the derivative of $\ln \lvert \csc x + \cot x \rvert$? [closed]

What is the derivative of $\ln \lvert \csc x + \cot x \rvert$? I've tried to do it and I get really odd numbers, any help showing the steps would be very helpful!
0
votes
0answers
10 views

Calculating vth moment of log-normal pdf describing particle size distribution

The number of density particles having radii between $r$ and $r+dr$, is given by the equation: $$n(r)=\frac{1}{\sqrt{2π} r \ln(σ_x)} e^{-\frac{ln^2(r/r_g)}{2ln^2(\sigma_x)}}$$ where $\sigma_x$ is ...
0
votes
2answers
25 views

Multi-valued logarithmic function

I'm reading some notes for an electrical engineering class and came to the following: "...$2^j$ can represent a countably infinite number of real numbers. These examples are related to the fact that ...
2
votes
0answers
22 views

Bias induced by splitting a log sum into independent log [closed]

Does anyone know the introduced bias ($\epsilon$) when a log-sum is split into a sum-log $$\log \left(\sum_{k=1}^N a_k\right) = \sum_{k=1}^N \log(a_k) + \epsilon$$ Many thanks for your help!
-2
votes
3answers
44 views

Differentiation using logarithms.

the variables $x$ and $y$ are positive and related by $$x^a\cdot y^b=(x+y)^{(a+b)}$$ where $a$ and $b$ are positive constants. By taking logarithms of both sides, show that ...
3
votes
1answer
95 views

Sum of infinite series and ratio of powers.

If $A = 1 + r^{a} + r^{2a} + \cdots ~~~~~ ;~~~ B= 1 + r^{b} + r^{2b} + \cdots $ Then $a/b$ is ? I took the sum of both series : $$A = \dfrac{1}{1-r^a}~~;~~~ B=\dfrac{1}{1-r^b}$$ But now how do we ...
7
votes
2answers
537 views

Is there ANY possible way to solve this equation?

So I came up with this equation and it just seems like I can't solve it AT ALL for '$a$' $$a*b^a = c$$ EDIT: By the way, I'm only taking $b^a$, not both $b$ and $a$, just in case anyone was ...
3
votes
6answers
127 views

Is there proof show that $\log x$ is undefined and make no sense at $ x=0$?

I was asked by someone: why $\log x$ is undefined at $x=0 $? Is there proof show that $\log x$ is undefined at $x=0$? Note(01):: log is the inverse function of the exponential function. note(02): ...
1
vote
3answers
66 views

How do I prove $\log(x^n)=n\log|x|$?

By definition we know that: $\log(x^n)=n\log|x|$ as known property in logarithm function . If it's not a trivial question, how do I prove that :$\log(x^n)=n\log|x|$? Note: $x$ is real number, $n$ is ...
1
vote
3answers
56 views

Solving the exponent function for X

Natural logarithm is defined as: $\ln(Y) = x$ Which can be also written as: $e^x = y$ Now the problem is, to solve the above equation for x you would need to use logarithm, unless the base can be ...
0
votes
2answers
43 views

I need help solving a logarithm equation.

these were my steps. Can someone tell me where have i goe wrong since the answer is $-{\frac 14}$. $\sqrt{\log(\sqrt{10}a)} = {\frac 12}$ ${\frac 12}{\log(\sqrt{10}a)} = \log\sqrt{10}$ ...
1
vote
2answers
19 views

Problem in substitution

I have a very stupid question, it seems that I've forgotten most of my math and can't figure this out. Considering the following, ...
0
votes
2answers
75 views

What is $\text{log}(-x)$?

I am having some confusion in regards to the log based value of a negative number. I know that this is said to be undefined, though I accidentally entered in '$\log(-x)$' instead of '$\log(x)$' via a ...
3
votes
1answer
55 views

$\ln r+\ln q=kr$ Isolating $r$

A problem I'm working on requires me to solve $\ln r+\ln q=kr$ for $r$. I've tried using the Lambert $W$ function, but I'm not sure how to do it. Is there method, technique or known solution, that ...
1
vote
1answer
52 views

Logarithm doubt …

I know that log of a negative number is not possible but, $\log(-5)^2$ is possible. Therefore $\log(-5)^2=2\log(-5)$ but $\log(-5)$ is not possible but $log$ of $-5$ square is possible ....can anyone ...
0
votes
1answer
49 views

Why is the function $\operatorname{Log}(G(t))$ Holder continuous?

I was reading the theory about the Riemann-Hilbert problem $\Phi^+(t)=G(t)\Phi^-(t)$ where $G(t)$ is a Holder continuous function on a closed curve $c$ with index $\operatorname{Ind}_cG(t)=0$. To ...
2
votes
4answers
354 views

Is there any way to prove this without logarithms?

I was given this problem: Show if $a>1$ and $n>1$ ($n$ and $a$ are integers) then, $\lim_{n\to\infty}a^{\frac{1}{n}}=1$ . The obvious solution is the following: Take the logarithm in base ...
0
votes
2answers
83 views

Transcendental numbers & logarithms

Given two coprime positive integers greater than one, say $n,\ m$ , where $n > m$ . How do we find the ratio $\dfrac{\log m}{\log n}$ in terms of $n$ and $m$ symbolically ? Claim: The ratio is ...
4
votes
1answer
47 views

Iteration of $\log(z) / \sqrt{z}$

The complex function $\log(z) / \sqrt{z}$ is a curiosity that I find interesting since one can express $e^{i\pi}+1=0$ as $\log(-1) / \sqrt{-1} = \pi$. My question is, what is the significance of the ...
0
votes
1answer
42 views

solve logarithmic equation without numerical methods

Is there algebraic method to solve following equation for $x$: $$ a x + b \ln x + c = 0 $$ with $a , b , c$ constants without using numerical methods and ln means natural logarithm.
0
votes
0answers
41 views

How to calculate $\log \log \log N$?

How to calculate $\log \log \log N$ effectively? Is this problem polynomial? I tried to solve this by my own, but I still have no results and ideas. I think there is a solution better than ...
0
votes
0answers
16 views

How to compute the logarithm of a computable number

Let's say you have a computable number $x>0$. $\ln(x)$ is computable as well. Given the computability of $x$, what is a computation for $\ln(x)$. I am using the definition where $a$ is computable ...
1
vote
2answers
42 views

Simplifying logarithm question

Without worrying about the background, I have a question that asks to solve for n. Pardon my formatting, but it seems understandable this way for the time being until I edit it: $$4n^2 = 256 ...
0
votes
0answers
7 views

negative sign in direction of wave propagation

Say I have a EM wave that goes in the Z direction and E=Eo*exp(-jkz). Why does the negative sign mean the wave travels in the +Z direction and exp(+jkz) means it travels in the -Z direction?
-2
votes
1answer
38 views

Convergente of serie log. [closed]

I would ask some help to solve the next set and how to explain their sum $$ \sum \limits^{\infty }_{n=2}\frac{\log[(1\text{+}\frac{1}{n} )^{n}(n+1)]}{\log(n)^{n}\log\text{(}n+1)^{n+1}} ...
2
votes
2answers
45 views

The minimum value of $\log_{10}x+\log_x 10$

Notation: $\log:=\log_{10}$ $\log x+\log_x 10$ $=\log x+ \frac{1}{\log x}$ $=\log(x \cdot \frac{1}{x})$ $=\log 1$ $=0$ Is the process correct? I doubt this is wrong. Please help. ...
0
votes
3answers
66 views

logarithmic and polynomial equation

I have the following $(1-a^x)/x=b$ Can this be solved for x ? (if yes, how, if not why) I have gotten to many forms, but can't seem to isolate x.
2
votes
4answers
53 views

Is showing $\lim_{z \to \infty} (1+\frac{1}{z})^z$ exists the same as $\lim_{n \to \infty} (1+1/n)^n$ exists

My expanded question: Is showing $\lim_{z \to \infty} (1+\frac{1}{z})^z$ exists as $z$ goes through real values the same as $\lim_{n \to \infty} (1+\frac{1}{n})^n$ exists as $n$ goes through ...
1
vote
1answer
34 views

Difference between the formula of Roger Cotes and Euler

What was the difference between the formula that Roger cotes derived and that euler got? I mean to say that Euler got the following formula : $$e^{ix} = \cos x+i \sin x$$ And Cotes got the following ...
0
votes
2answers
34 views

A Simple Logarithm Question

Solve for $x$: $\log_2 (2x+8)=3$ Correct Solution: $2x+8=2^3$ $2x+8=8$ $2x=0$ $x=0$ Why doesn't this work: $\log_2 (2x+8)=3$ Expand: $\log_2(2x)+\log_28=3$ $\log_2(2x)+3=3$ $\log_2(2x)=0$ ...
0
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2answers
59 views

Is $\sqrt{\log (n)}=\frac{1}{\sqrt{2}}*(\log n)$? [closed]

Is $$\sqrt{\log (n)}=\frac{1}{\sqrt{2}}*(\log n)$$
0
votes
3answers
60 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 ...
0
votes
1answer
37 views

Solve by separation of variables: $\frac{dx}{dy}y\ln|x| = \big(\frac{y+1}{x}\big)^2$

I need to solve the problem above using separation of variables. I got as far as the below but it seems too complex to be right. Am I wrong somewhere? Because I think my final answer needs to simplify ...
3
votes
3answers
83 views

Find sum of series [closed]

I need to find the sum of the following series: $$\sum_{n=2}^\infty \ln\left(1-\frac 1{n^2}\right)$$ How to proceed with this?
0
votes
2answers
31 views

Sequences identity

I have some problems to find a way to prove the following statement, if someone could give me any suggestions would be grateful: Show that $$ log\text{(}a_{n}+\text{1})\approx a_{n} $$ when $$ ...
4
votes
2answers
245 views

If $x$ is rational, can $\log(1-x)/\log x$ be algebraic?

If $x$ is positive rational number less than $\frac{1}{2}$, can the following logarithmic expression be equivalent to an algebraic number, say $g$? $$\frac{\log(1-x)}{\log x} = g$$
1
vote
2answers
61 views

Evaluate $\log_{2005}(1/2)\log_{2004}(1/3)\log_{2003}(1/4)\ldots\log_2(1/2005)$ [closed]

The numbers 2005, 2004, 2003, ..., 2 are the bases. I cannot understand how to start the question. Please help. What to do in these type of questions? Thanks in advance.
0
votes
2answers
21 views

Logarithm where $0<a<\frac{1}{2}$. Find $x$

Given that $\log_a(3x-4a)+\log_a(3x)=\frac{2}{\log_2a}+\log_a(1-2a)$ where $0<a<\frac{1}{2}$. find the value of $x$. I got the attempt until $x=\frac{2(a+\sqrt{(a-1)^2}}{3}$ and ...
2
votes
1answer
49 views

Are these proofs of the 1st and 3rd Laws of Logarithms valid?

Disclaimer: I dont mean that I've discovered a conceptually completely different way of proving those laws, of course. I just found myself proving them like this and then realized that they're ...
1
vote
1answer
53 views

For which values of a parameter an equation has one Real root

The following equation is given $$\log_{x-1}(x^2+2ax) - \log_{x-1}(8x-6a-3)=0$$ And I am trying to find for which values of $a$ it has only one root, which is real. It is obvious that $$x-1>0 ...
1
vote
0answers
35 views

How do I show that the integral $\int_0^\infty x^{-a} |\log x|^b dx$ only converges when $a = 1$ and $-2 < b < -1$?

This came up in a previous question, but was closed because the question wasn't terribly clear. I don't want to edit the other question substantially because it's not mine so I'm asking a new one and ...