Questions related to real and complex logarithms.

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0
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2answers
94 views

How have they simplified this function?

I have been trying to figure out how the following has been simplified, but I am getting nowhere with it. Anyone have any ideas? $9(n/3)^{5/2}$ to $(1/3)^{1/2} f(n)$ It is given that $f(n) = ...
2
votes
0answers
82 views

Is there a (hopefully free) graphing utility BESIDES mathematica that can graph log polar coordinates?

This is my first post to the forum. I have had a good experience with other stackexchange fora, so I have high hopes for this one. I have looked online and can not seem to find the software described ...
0
votes
1answer
45 views

Logarithm exponent in Chernoff bound

I am applying Chernoff bound for a Poisson process with mean $\lg n$. I am putting $\delta =4$. Hence, $Pr(X<(1+4)\mu)< (\frac{e^\delta}{(1+4)^{(1+4)}})^\mu$ $ = (\frac{e^\delta}{5^5})^{\lg ...
0
votes
2answers
97 views

Proof of $\log_{b+c} {a}+ \log_{c-b}{a}= 2\log_{b+c} {a} \cdot \log_{c-b}{a}$ when $a^2=b^2+c^2 $

If $ a^2=b^2+c^2 $ prove that $$\log_{b+c} {a}+ \log_{c-b}{a}= 2\log_{b+c} {a} \cdot \log_{c-b}{a}.$$ thanks
4
votes
2answers
350 views

How to calculate number of digits of a large number?

Does anyone know any efficient ways of finding the number of digits in the large number $N = 4^{4^{4^4}}$? Thanks.
2
votes
1answer
92 views

Solving $ f(\log x)$

A generalization of the conjecture $$\pi(x+x^{\theta}) - \pi(x) \sim \frac{x^\theta}{\log x} $$ (Ingham, 1937 or earlier) might be $$\Delta \pi_k = \pi_k((x+1)^2) - \pi_k(x^2)\sim \frac{x}{\log ...
6
votes
3answers
184 views

Is $ \log_a x^n = n \log_a x? $

We have: $$\log_a{x^n}=\log_a{\left( x\cdot x \cdot x \cdot ...\cdot x\right)}=\log_a(x)+\log_a(x)+...+\log_a(x)=n\log_a(x)$$ But, $$ \ln\left(-1\right)^2=\ln{\left(-1\right)^2}\\\ln(1)=2\ln(-1)\\ ...
6
votes
4answers
203 views

Proof of $\log_xy=\frac{\log_zy}{\log_zx}$

Why is $\log_xy=\frac{\log_zy}{\log_zx}$? Can we prove this using the laws of exponents?
1
vote
2answers
122 views

Why are expressions such as $\operatorname{ln}(T)$ used in thermodynamics where $T$ is not dimensionless?

In all thermodynamics texts that I have seen, expressions such as $\operatorname{ln}T$ and $\operatorname{ln}S$ are used, where $T$ is temperature and $S$ is entropy, and also with other thermodynamic ...
4
votes
3answers
534 views

Solving $xe^{-x}+2e^{-x}=0$

While I was studying my maths book, I came across this equation: $$ xe^{-x}+2e^{-x}=0 $$ I tried to solve it in different ways, but each time I break up some rule. My best try was this: Let's ...
1
vote
5answers
418 views

L'Hospital's Rule Question.

show that if $x $ is an element of $\mathbb R$ then $$\lim_{n\to\infty} \left(1 + \frac xn\right)^n = e^x $$ (HINT: Take logs and use L'Hospital's Rule) i'm not too sure how to go about answer this ...
14
votes
3answers
336 views

Find $x$ from $3^x\cdot x^3 = 1$

I saw a question on internet, tried to solve but I can't: \begin{equation} 3^x\cdot x^3 = 1 \end{equation} I get $\ln$ function and made some equalization and I reached that: \begin{equation} ...
1
vote
4answers
204 views

Derivative of compositum function with log

I have the following two functions that I'm not compleately sure I'm solving correctly mainly what bugs me is $\log(x)$. 1st Function: $$ f(x) = \sin(2x^2 - 3\log(x)) $$ I simply treated this as ...
2
votes
1answer
487 views

Deriving a power equation from a log-log line equation

I have a log-log plot of my data (see below) The equation of the line was determined to be: $5.26 + x0.7089$ If I wanted to convert this into a power equation would the correct way be: $ ln(y) = ...
16
votes
4answers
472 views

How to calculate $I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$?

How do I integrate this guy? I've been stuck on this for hours.. $$I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$$
3
votes
2answers
116 views

Evaluate: $\lim_{n \to \infty}[(1+\frac{1}{n})^n-(1+\frac{1}{n})]^{-n}$

Evaluate: $$\lim_{n \to \infty}[(1+\frac{1}{n})^n-(1+\frac{1}{n})]^{-n}$$ attemp: Take $P=\lim_{n \to \infty}[(1+\frac{1}{n})^n-(1+\frac{1}{n})]^{-n}$ . Then taking log both side .$$\ln ...
5
votes
4answers
290 views

Can we prove $a^{\log_bn} = n^{\log_ba}$?

Can we prove $$a^{\log_bn} = n^{\log_ba}?$$ I forget how to prove this theorem. I picked up one numbers for test, and they worked.
1
vote
1answer
192 views

Finding the Function to a log-log plot

My maths is not very good, so please bare with me if the answer is obvious. I have a log-log plot below, which I have generated in R. I now need to find what i believe to be called the inverse to ...
0
votes
1answer
28 views

How to limit the upper and lower sizes of these resulting log outputs?

I've got quite a puzzle for ya mathematicians here :) This could go in the regular stackoverflow but since the subject matter of the question is primarily math I figure I'd ask it here. I wasn't ...
7
votes
4answers
2k views

Prove that if $a^x=b^y=(ab)^{xy}$, then $x+y=1$.

Prove that if $a^x=b^y=(ab)^{xy}$, then $x+y=1$. How do I use logarithms to approach this problem? Any help would be appreciated, thanks.
3
votes
1answer
100 views

Repeated logarithm

By definition $\ln = \log_e$ on complex numbers is given by $$ \ln(re^{i\theta}) = \ln(r) + i\theta $$ $(-\pi < \theta\leq \pi, r >0)$. Then $\ln(-1) = \pi i$. And $\ln(\pi i) = \ln(\pi) + ...
2
votes
1answer
422 views

Natural log summation representation

In working through a problem, I've encountered the need to express $\log n = \sum \limits_{k = 1}^{n - 1} \log(1 + \frac{1}{k})$ where $\log k $ is the natural logarithm of k. I'm fairly certain the ...
3
votes
1answer
172 views

What are the asymptotics of the solution to $\log x=\epsilon x$?

I just read the question Why does $\ln(x) = \epsilon x$ have 2 solutions?, and thought I'd point out a related area of investigation. The equation $\log x=\epsilon x$ has 2 solutions for ...
1
vote
2answers
1k views

The logarithm of 3 base 10 is irrational

Prove that the logarithm of 3 base 10 is irrational The Fundamental Theorem of Arithmetic is that every integer is a product of primes. So far I have, Suppose $\log_{10}(5)$ is rational. Then ...
1
vote
5answers
129 views

Solution for $\log_7x+\log_{\frac17}x^2=\log_{49}x-3$

What is the right solution for $\log_7x+\log_{\frac17}x^2=\log_{49}x-3$. What logarithm identities used?
1
vote
0answers
77 views

Use of natural logarithm transformation on weighted index series

I have a value computed as sum of powers, e.g. $x^5+y^8+z^2$. The exponent represents the weight for variables, $x, y$ and $z$ in the example above. Applying natural logarithm on $x^5+y^8+z^2$, I get ...
0
votes
4answers
342 views

Prove that the limit $\displaystyle\lim_{x\to \infty} \dfrac{\log(x)}{x} = 0$

Prove $$\lim_{x\to \infty} \dfrac{\log(x)}{x} = 0$$ and $$\lim_{x\to \infty} \dfrac{\log(x)}{x^n} = 0$$ From the definition of $\log(x)$, $$\log(x) = \int_1^x \dfrac{1}{t} dt$$ Since $1$ ...
4
votes
7answers
664 views

Prove that $5/2 < e < 3$?

Prove that $$\dfrac{5}{2} < e < 3$$ By the definition of $\log$ and $\exp$, $$1 = \log(e) = \int_1^e \dfrac{1}{t} dt$$ where $e = \exp(1)$. So given that $e$ is unknown, how could I ...
1
vote
1answer
85 views

Could you describe this function as “logarithmic”?

Consider the following function: $$f(x) = \frac{1}{\sqrt{x}}$$ As $x$ increases, the value of $f(x)$ decreases, but the decrease tapers off quickly as $x$ gets larger, and if you plot the graph of ...
1
vote
1answer
24 views

Sign when taking logarithm

$$F=\left({\frac 1\theta}\right)^n\cdot e^{\sum_1^n x_i/\theta}$$ Taking logarithm, get $$-n\ln\theta-\sum_1^n x_i/\theta$$ Why do we get a minus sign in the second expression before the summation?
3
votes
3answers
307 views

Solve $\log x + \log(2x-5) = \log 96 – \log 8$

Solve $\log x + \log (2x-5) = \log 96 – \log 8$. I started by doing this: $\log 2x^2-5x = \log 12$ Then I got rid of the logs. Can I do that? $2x^2-5x = 12$ What do I do next? What is $x$?
2
votes
2answers
150 views

changing equation to logarithmic form and solving it

$$3^{x+1} = 3000$$ How do I solve this? I know we use logarithms but I do not remember how to solve this kind of problem. I am guessing that I need to change the problem into log form. but how? ...
0
votes
1answer
82 views

Logarithm, LUT and how to make it non-linear

Currently I calculate log as following: ...
0
votes
1answer
104 views

Vernier scale on logarithmic scale

I know that a vernier scale can be used to accurately read a linear scale, such as in vernier calipers. I'm wondering if there is a way the methods behind a vernier scale could be adapted for usage ...
1
vote
3answers
62 views

Solve the equation $22\log(92x+40.66)=38.9$

The equation $$22\log(92x+40.66)=38.9$$ steps so far $$\log(92x+40.66)=\frac{38.9}{22}$$ to eliminate log, do I have to apply the opposite of log? Not sure what that is.
2
votes
2answers
81 views

Explanation of this inequality

Is there a graphic visualization of $\sum_{k=1}^{n} 1/k \, \, \leq \, \, \,1 \, + \, \int_1^n \! (1/x) \, \mathrm{d} x$ as intuitive as the integral test ? I can't see why the inequality is true. I ...
2
votes
1answer
106 views

Solve the equation using logarithms

Equation $$ e^{2x+1.21} = 114\cdot 4^x $$ steps I've done so far. $2x + 1.21 = \ln(114) \cdot \ln(4) \cdot x$ $x = (\ln(114) * \ln(4))/1.21$ I don't think I was allowed to move the $x$ from the ...
2
votes
3answers
90 views

Solve $33(1.207)^x = 47(1.547)^x$

How can we solve this equation? $$33(1.207)^x = 47(1.547)^x$$ I've done this step so far.$$33x\ln(1.207) = 47x\ln(1.547)$$ This is where I get stuck. Won't moving an $x$ to one side eliminate all ...
2
votes
1answer
66 views

Solve the equation by using logarithms

The equation is $64(12)^{8x} = 195$ the steps I have done so far. $12^{8x} = 195/64$ $8x \cdot \ln(12) = \ln(195/64)$ $8x = (\ln(195/64))/(\ln(12))$ not sure how to divide $8x$ to get $x$ by ...
3
votes
3answers
93 views

How to solve problems such as $x = \log_2{x}$

How does one go about solving $x = \log_2{x}$? Is there a technique to solve these sorts of problems?
5
votes
2answers
410 views

Prove that $\log(1 + \sqrt{1+x^2})$ is uniformly continuous?

Problem Prove that $$\log(1 + \sqrt{1+x^2})$$ is uniformly continuous. My idea is to consider $|x - y| < \delta$, then show that $$|\log(1 + \sqrt{1+x^2}) - \log(1 + \sqrt{1+y^2})| = ...
0
votes
1answer
88 views

Can $2^n$ + $2^m$ be expressed as $2^x$

Can $2^n$ + $2^m$ be expressed as $2^x$ where $x$ is a function of $n$ & $m$? I'm sure that this would require logarithms to find the answer but my maths is very rusty. Can anyone point me at the ...
1
vote
1answer
307 views

Base 2 logarithm with Taylor expansion

I'm trying to implement the natural logarithm in C, and our task is to make it really efficient. So what we are doing is, that we use the first 8 members of the series. This works fine, but the ...
2
votes
3answers
1k views

Taylor series expansion of base 2 logarithms

Sorry for the noob question, but I've been hitting my head against the wall on this for a while. I am looking for a Taylor series expansion of a logarithm other than the natural logarithm $ln(x)$. It ...
6
votes
3answers
204 views

What operations on an equation cause it to be destroyed?

I approached my calculus professor about something he said which didn't make much sense to me - He says that in the process of calculating $\lim_{x\to\infty} f(x)^{g(x)}$, you can convert it to ...
3
votes
1answer
233 views

Understanding accuracy of Newton's Method

In a numerical analysis book I'm reading it says that using the Newton error formula we can find an expression for the number of correct digits in an approximation using Newton's Method. Here's the ...
2
votes
2answers
195 views

Prove $\lfloor \log_2(n) \rfloor + 1 = \lceil \log_2(n+1) \rceil $

This is a question a lecturer gave me. I'm more than willing to come up with the answer. But I feel I'm missing something in logs. I know the rules, $\log(ab) = \log(a) + \log(b)$ but that's all I ...
0
votes
2answers
109 views

Differentiation Of Natural Logarithms

The problem I have is to differentiate $ y = ln(x^4)$ Using the rule : $$\frac{d[lnf(x)]}{dx}=\frac{f'(x)}{f(x)}$$ My working is: $$\frac{dy}{dx} = \frac{x^4ln(x)}{x^4}$$ $$=ln(x)$$ but the book is ...
1
vote
1answer
77 views

What kind of series / recursion is this?

I'm trying to find the explicit solution / sum of first n elements for the following sequence: d(2) = 2 d(n) = d(n/2) + n*log2(n) Can you help me to find out ...
9
votes
9answers
627 views

Show that, for $t>0$, $\log t$ is not a polynomial.

How can I show that? I've tried to reverse the logarithm to it's exponential form in a trial to show that but I got no success. Can you help me?