Questions related to real and complex logarithms.

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-1
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1answer
85 views

Find a relation between and y that does not involve logarithms

Could I please have a solution to this, I've spent an hour on it so far -_- Thanks in advance. $$ \log_{10}(1+y) - \log_{10}( 1-y) = x$$
1
vote
0answers
45 views

$f(n) = n^2 \lceil \log n \rceil$ is time constructible

I have a question, I want to show, that: $$f(n) = n^2 \lceil \log n \rceil $$ is time-constructible. I have no idea how to do this. I know that $n^2$ is time-constructible and I know that $\log n$ ...
4
votes
2answers
236 views

Limit of logarithms without l'Hospital

This is my first post so I hope you forgive any formatting mistakes. This is a task out of a training exam, I may add that we have not yet introduced l'Hospital or derivatives. We have to determine ...
1
vote
1answer
992 views

Another two hard integrals

Evaluate : $$\begin{align} & \int_{0}^{\frac{\pi }{2}}{\frac{{{\ln }^{2}}\left( 2\cos x \right)}{{{\ln }^{2}}\left( 2\cos x \right)+{{x}^{2}}}}\text{d}x \\ & \int_{0}^{1}{\frac{\arctan ...
0
votes
2answers
56 views

For what $f(n)$ does $O(f(n) \log n)=O(\log\log n)$?

$k=f(n)$. Given $O(k \log_2 n)$, what function $f$ of $n$ would be needed for it to equal $O(\log_2 \log_2 n)$? (where $k \in n \in \mathbb{Z}^+$)
2
votes
4answers
209 views

Solving $\;5^{2x}-4\cdot 5^x=12$

I need to solve $\quad\displaystyle 5^{2x}-4\cdot 5^x=12$. I've only gotten this far: $\quad \displaystyle 5^{2x}-20^x=12.$ I don't know what to do next. Thanks in advance!
2
votes
2answers
121 views

About the use of Stirling approximation

How to prove this inequality: $$\ln \Gamma \left( x \right)-2\ln \Gamma \left( \frac{x+1}{2} \right)>\frac{2x}{3}$$ Sry I forgot to mention that $x>300$
2
votes
1answer
104 views

Some log and exponential integral

I think these should hav some closed form: $$\displaystyle\begin{align*} & \int_{0}^{1}{\frac{\left( 1-x \right)\ln \left( x \right){{\text{e}}^{-x}}}{\pi -x}}\text{d}x \\ & ...
3
votes
3answers
561 views

Solving inequality involving logarithms

I must be doing something wrong. I want to solve the following, where n is a positive integer, and p is a real number between 0 and 1. $$(1-p)^n \le 0.4$$ So I take the log on both sides: ...
2
votes
5answers
543 views

Convergence of series $\sum_{n=1}^\infty \ln\left(\frac{2n+7}{2n+1}\right)$?

I have the series $$\sum\limits_{n=1}^\infty \ln\left(\frac{2n+7}{2n+1}\right)$$ I'm trying to find if the sequence converges and if so, find its sum. I have done the ratio and root test but It ...
1
vote
2answers
740 views

Log laws and modulus

If you have the log of a modulus, (like after integration), how do the log laws work? So if you have $a\ln\left|2x-3\right|$ does it become: $\ln\left|(2x-3)^a\right|$ or $\ln(\left|2x-3\right|)^a$, ...
2
votes
1answer
124 views

Two log trig integral

$$\begin{align*} & \int_{0}^{\frac{\pi }{2}}{{{\ln }^{n}}\sin x\text{d}x} \\ & \int_{\frac{\pi }{4}}^{\frac{\pi }{2}}{\ln \left( \ln \tan x \right)}\text{d}x \\ \end{align*}$$
3
votes
3answers
176 views

How can I show that $f(x) = (x^2)/(1-e^x)$ has global minimum at $(0, +\infty)$?

I showed that $\lim f(x) = 0$ at both the $0$ end and $+\infty$ end. What is the proper way to finish the proof?
0
votes
0answers
58 views

What does taking the logarithm of a variable mean?

Question with regards to taking the logarithm of a variable (Statistics Question) Say you have a bar graph displaying data for an example "Cost of Computer Orders by the Population" and you are ...
0
votes
1answer
82 views

Making data fit a certain equation form?

I have a set of data that follows a polynomial (parabola) equation almost perfectly, but I need it to be of the form $T(n) = a\cdot n + b\log(n)$ I approximated it to the polynomial equation using ...
4
votes
1answer
166 views

Solve $8 \log(x) - x = 0$

Someone came to me recently with this seemingly simple equation to solve: $$8 \log(x) - x = 0$$ So far, everything I have tried has been a dead end. Is there a symbolic solution to this kind of ...
1
vote
1answer
90 views

$x \sim y \implies \pi(x) \sim \pi(y) $ and repeated applications of PNT

Let $\sim$ mean if $a \sim b$ then $\lim_{x \to \infty} \frac{a}{b} =1.$ The following is a threshold question. It seems that $x \sim y \implies \pi(x) \sim \pi(y).$ Pf. $\pi(x) \sim \frac{x}{\log ...
3
votes
3answers
159 views

$x \sim y \implies \log x \sim \log y$?

Does $x \sim y \implies \ln x \sim \ln y$? I would have thought not, but the following has almost persuaded me otherwise: Assume $x \sim y.$ Does this imply that $$\tag{1}I = ...
3
votes
1answer
258 views

exponential population growth models using $e$?

Im trying to understand this write up [1] of cell population growth models and am confused about the use of natural logarithms. If cells double at a constant rate starting from 1 cell, then their cell ...
5
votes
3answers
209 views

Why is $\int\limits_{1}^{n} \log x \,dx \le \sum\limits_{x = 1}^{n}\log x$?

It has been a long time since I studied integrals, so this question may sound stupid. I was going through this wiki page, and came across the following inequality: $$\int_{1}^{n} \log x \,dx \le ...
1
vote
1answer
235 views

log-odds to probabilities

For example the variance in the log-odds is $0.07$ $(0.01)$, and the mean log-odds is $-0.65$ $(0.03)$. Just transforming this variance to probability given me $0.51$. So if the probabilities have a ...
1
vote
0answers
48 views

Calculating the number of states in power of 10's

This is part of a problem that was assigned to me. It might seem elementary, but I need a hint on how to start this problem. I have 10 billion bits that can either be on or off at any time. ...
1
vote
2answers
88 views

Derivative for log

I have the following problem: $$ \log \bigg( \frac{x+3}{4-x} \bigg) $$ I need to graph the following function so I will need a starting point, roots, zeros, stationary points, inflection points ...
0
votes
1answer
133 views

Checking whether answers of logarithmic and exponential equalities are correct.

When you check the answers you get from equalities like for example: $$ ^2\log(x-2) = 3- ^2\log(x)$$ $$ 4^x = 3 \times 2^x + 10$$ so on and so forth, is it sufficient to do the following: For the ...
1
vote
1answer
69 views

Trying to convert a nasty logarithm into an exponential

I have the following equation that I must express in terms of $r$: $$\Delta V = \frac{\lambda}{2 \pi \epsilon_0} \ln(\frac{r}{R})$$ This is a pretty tough one. I am not sure how to get the r out of ...
2
votes
3answers
74 views

Shouldn't these 2 be equivalent?

$ 2 \ln (5x) = 16$ $ \ln (5x) = 8 $ $ 5x = e^8 $ $ x = \dfrac {1}{5}e^8$ But why can't we do it like this: $ \ln(5x)^2 = 16$ I thought that was a possibilty with logaritms?
1
vote
3answers
91 views

What is wrong with this solution?

$$ \ln(x) = 1 + \ln(5)$$ $$ x = e^{1+ \ln(5)} = e^{1+5} = e^6$$ What exactly am I doing wrong here?
2
votes
1answer
20 views

Estimation for large $k$.

I have a function $f(k)$ defined on the set of natural numbers and I managed to show that $f(k)>n-\binom n k(1-n^{-2/3})^{k(k-1)/2}$ for all integers $n\ge k$. I am hoping to get a further ...
0
votes
1answer
88 views

Is $\lim_{x \to x_0} \log(f(x)) = \log\lim_{x \to x_0} f(x)$ always true?

This property is always true? If yes I would like a proof, otherwise an counterexample. $$\lim\limits_{x \to x_0} \log(f(x)) = \log\lim\limits_{x \to x_0} f(x)$$
4
votes
2answers
592 views

Logarithm as limit

Wolfram's website lists this as a limit representation of the natural log: $$\ln{z} = \lim_{\omega \to \infty} \omega(z^{1/\omega} - 1)$$ Is there a quick proof of this? Thanks
3
votes
2answers
2k views

What does lg x mean? is it $\log_2 x$ or $\log_{10} x$ in binary trees

I'm a bit confused, $\log_{10} x = \log x $ right? I believe I've read somewhere that $\log_{2} x = lg x$ but some people say lg = $\log$. So what does lg really stand for? specifically when talking ...
0
votes
1answer
41 views

Bounding below the difference of sums

I would like to bound below the following expression: $\lambda(m,n)=\sum\limits_{i = 1}^{m+n}\lg{i} - \sum\limits_{i = 1}^{m}\lg{i} - \sum\limits_{i = 1}^{n}\lg{i}$ by some expression that ideally ...
1
vote
0answers
41 views

Log return of two different timeseries

Lets say I have a single timeseries, the simple return would be T/T-1-1 the log return would be ln(T/T-1) But let's say I have two different time series, T and R The values are close, but still ...
1
vote
1answer
116 views

Given the value of $\log n$, what is the value of $\log 2n$?

I had the attached multiple choice question in an online practice exam. I'm not sure how to work this out.
1
vote
3answers
114 views

Equation involving logarithm, solvable without calculator?

I'd like to know if I can solve the following equation without calculator: $(0.4)^t=5t$ I don't think it's possible, cause I always get stuck on formulas of the form $e^t=t$ or $t=\ln t$ I've also ...
0
votes
2answers
84 views

Explanation needed on this rather basic recurrence solution

We are studying about recurrences in our analysis of algorithms class. As an example of the substitution method (with induction) we are given the following: $$T(n) = \lbrace 2T\left(\frac{n}{2}\right) ...
1
vote
1answer
44 views

Horizontal line segments and logarithmic functions

We have the functions $$f(x)= \log_2(x+3), \quad\text{ and }\quad g(x)= 1 + \log_{1/2}(x)$$ Find for which values of $q$ the graphs cut of a line segment of $2$ of the line $y=q$. (because of my ...
3
votes
3answers
103 views

Solving equation that involves nepper number

I have the following equation. $$\frac{1}{5} = \large e^{\frac{10^{-4}}{x}}$$ I assume that x will be $0.0621$ but I can't remember how does one solves such equation. I recall that it involves using ...
3
votes
3answers
7k views

The relation between an exponential function and a logarithmic function

I have been told multiple times that the logarithmic function is the inverse of the exponential function and vice versa. My question is; what are the implications of this? How can we see that they're ...
1
vote
2answers
200 views

Question regarding differentiating logarithmic functions

I have a couple of small questions regarding differentiating logarithmic functions: The derivative of $ \log(x)^2 = \dfrac{2}{\ln(10)x}$ The derivative of $ 2 \log(x) = \dfrac{2}{\ln(10)x}$ Does ...
1
vote
2answers
320 views

Logarithm question for Algebra 2/Trig class

$$\frac{1}{2} \log(x+2)=2$$ I'm decently good at logarithms but this one seems to be tricky, when I did it myself I got a negative decimal as my answer but I'm not 100% confident in it, and I would ...
8
votes
5answers
241 views

Solving an equation with a logarithm in the exponent

I try to solve the following equation: $$ (N+1)^{\log_N{125}} = 216 $$ I know the answer is 5 here but how could I rewrite the equations so I can solve it? I tried to take the log of both sides but ...
4
votes
2answers
193 views

I am trying to solve the inequality $\log_{\log{\sqrt{9-x^2}}} x^2 <0$

I am trying to solve the inequality $$\log_{\log{\sqrt{9-x^2}}} x^2 <0.$$ I got $\mathrm{S.S}=(-\sqrt8 ,-1)\cup( 1,\sqrt8)$, but a friend got $\mathrm{S.S}=(-1,1)- \{0\}$. Please, what is ...
0
votes
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
86 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
48 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
99 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
377 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
93 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
186 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)\\ ...