Questions related to real and complex logarithms.

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7
votes
2answers
555 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
132 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
57 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 ...
1
vote
3answers
70 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 ...
0
votes
1answer
51 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 ...
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
77 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
56 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
54 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 ...
2
votes
4answers
366 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
117 views

Integration of 1/x as a limit of a sum

This is from R.Courant book Example "Introduction to Calculus and Analysis vol.1 " To integrate $x^\alpha$ when $\alpha\neq1$ we subdivide the interval [a,b] by the point of geometric progression: ...
4
votes
1answer
49 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
2answers
88 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 ...
0
votes
1answer
50 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
46 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
43 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
3answers
68 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.
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
2answers
46 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. ...
4
votes
2answers
251 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 $g$? $$\frac{\log(1-x)}{\log x} = g$$
2
votes
4answers
55 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
36 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: $\log_2(2x)+\log_28=3$ $\log_2(2x)+3=3$ $\log_2(2x)=0$ ...
7
votes
7answers
2k views

An alternative way to calculate $\log(x)$

How can I replace the $\log(x)$ function by simple math operators like $+,-,\div$, and $\times$? I am writing a computer code and I must use $\log(x)$ in it. However, the technology I am using does ...
0
votes
2answers
63 views
14
votes
2answers
288 views

Integral ${\large\int}_0^1\left(-\frac{\operatorname{li} x}x\right)^adx$

Let $\operatorname{li} x$ denote the logarithmic integral $$\operatorname{li} x=\int_0^x\frac{dt}{\ln t}.$$ Consider the following parameterized integral: $$I(a)=\int_0^1\left(-\frac{\operatorname{li} ...
0
votes
1answer
49 views

What role does $1/\alpha$ play in the last integral?

If we examine the inverse function $f^{-1}=log_{10}$, the whole situation appears in a new light: $$\begin{align} log_{10}'(x)&=\frac1{f'\left(f^{-1}(x)\right)}\\ &=\frac1{\alpha\cdot ...
0
votes
1answer
40 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 $$ ...
1
vote
2answers
64 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
2k views

Graphing: Given two points on a graph, find the logarithmic function that passes through both.

Is there such a method to do this? I would like to come up with a logarithmic function (a graph that looks like a square root graph) that passes through two given points. Haven't had any luck in ...
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 ...
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 ...
5
votes
4answers
3k views

log base 1 of 1

What is $\log(1)$ to the base of $1$? My teacher says it is $1$. I beg to differ, I think it can be all real numbers! i.e., $1^x = 1$, where $x\in \mathbb{R}$. So I was wondering where I have gone ...
5
votes
4answers
467 views

Proof $e^x = \exp(x)$?

Define $$\ln (x) = \int^{x}_{1}\frac{1}{t}$$ Assume I have proven that $\ln x$ is one-to-one and therefore has an inverse $\exp (x)$. Define $e$ as: $\ln e = 1$ Now, if you have no other notion ...
7
votes
2answers
2k views

How many digits does $2^{1000}$ contain?

I tried this way, I only need to know if this is correct or if there are better ways to solve this: $2^{1000}$ does not have a factor of $5$ obviously therefore we can assume $$ 10^{m} < 2^{1000} ...
1
vote
0answers
36 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 ...
2
votes
1answer
46 views

How to understand or derive the formula $Mantissa$ $bits$ $/$ ${log_2}$ $10$ = $Decimal$ $digits$ $of$ $precision$?

I asked a question a couple days ago about floating point precision on stackoverflow called, "Is floating point precision mutable or invariant?" and I received the following response. The formula ...
4
votes
5answers
54 views

Solving logarithmic equations including x

Let $$\log_3(x-2) = 6 - x$$ It's obvious drawing the graphs of the two functions that the only solution is $x=5$. But this is not really a proof, rather than observation. How do you prove it ...
3
votes
2answers
57 views

Regularizing the $\log\log n$ series

The divergent series $$\sum_{n=1}^\infty\log n$$ can be regularized using the derivative of the Riemann zeta function at $s=0$: ...
2
votes
3answers
42 views

Logs rules and Solving

I've got the equation : $$-1=\frac{-8e^{-t} + 3e^t}{2e^t}$$ I've moved some stuff around to get : $5e^t = 8e^{-t} $ But not sure where to go from here. Thanks for any help
2
votes
3answers
42 views

Set of real $a$ so that the inequality is defined but isn't true for a real $x$

$$x(x-\sqrt {4+\log_a7})\lt \log_7 \frac a{49}$$ I reach the interval $(0,1)$ after looking for the discriminant of the quadratic to be less than zero. However, the solution in the book is an ...
1
vote
1answer
19 views

Find distribution mean from the mean and sd of the log

I have a distribution with a long tail and use a model to predict the mean and standard deviation of its log. Given the mean and standard deviation of the log, how do I find the mean of the actual ...
1
vote
1answer
59 views

Derivative of $(\ln x)^e$ [duplicate]

In Randall Munroe's What If, he says that "if you want to be mean to first-year calculus students, you can ask them to take the derivative of $(lnx)^e$" He says, as I would expect, that the result ...
19
votes
4answers
4k views

“What if” math joke: the derivative of $\ln(x)^e$

Randall Munroe, the creator of xkcd in his latest book What if writes (p. 175) that the mathematical analog of the phrase "knock me over with a feather" is seeing the expression $ \ln( x )^{e}$. And ...
2
votes
1answer
33 views

Confused with integral and natural logarithm

When reading about ideal gas and adiabatic expansion, I got stuck with the following: $$W_{ab}=\int_{{\it V_a}}^{{\it V_b}}\!\,{\rm ...