Questions related to real and complex logarithms.

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-1
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1answer
26 views

why does $\frac{d}{dx} log_b(x)$ not = $\frac{lnb}{x}$?

I know that $log_b(x) = \frac{lnx}{lnb}$, and that differentiating $$\frac{d}{dx}(\frac{lnx}{lnb}) = \frac{1}{lnb}\frac{d}{dx}(lnx)=\frac{1}{xlnb}$$, so where is my mistake when I do it this way: ...
1
vote
1answer
18 views

How can I compute fast the minimum of a linear plus Kulback-Leibler on the unit simplex?

Given $a, x^0 \in \mathbb{R}^n$ I wish to compute $$\min_{x \in \Delta_n} a^t x + \sum_{i=1}^n x_i\log(x_i/x^0_i) - x_i +x^0_i $$ where $\Delta_n$ is the unit simplex $\{x \in \mathbb{R}^n \mid ...
5
votes
1answer
313 views

inequality $10<2^{2^{\frac {3}{\log_2 \log_2 10}}}$

While working on this question I ended up with $10<2^2{^{\frac {3}{\log_2 \log_2 10}}}$ I am looking for answers using methods similar to this or this or this or this. Alternative original ...
1
vote
2answers
23 views

Determining whether $f(z)=\ln r + i\theta$ (with domain $\{z:r\gt , 0\lt \theta \lt 2\pi\}$) is analytic [duplicate]

Define $$f(z)=\ln r + i\theta$$ on the domain $\{z:r\gt , 0\lt \theta \lt 2\pi\}$. This domain is just a punctured disk of radius $\ln r$, correct? How does one determine whether this is ...
0
votes
0answers
16 views

Find the set of points on which the maps of $e^z$ and $\log(z-1)$ are expanding and contracting.

I understand that $e^z$ is has a domain $\Omega$ such that $\Omega = \Bbb {C}$ and is analytic on the whole complex plane, but I have never been tasked with understanding the map of a function that is ...
-1
votes
1answer
28 views
1
vote
1answer
18 views

Initial value problem through origin

$\frac{dz}{dt}=8t*e^z$, Through the origin I have never done an initial value problem before, but I took it to mean that it gave me the initial value of the differential equation (0, 0) and that I ...
16
votes
3answers
206 views

How to prove $\sqrt[\pi]{e} < \sqrt[\pi]{\pi}<\sqrt[e]{e}< \sqrt[e]{\pi}$

I was given a challenge of sorting the following numbers. $\Large\sqrt[\pi]{e} < \sqrt[\pi]{\pi}<\sqrt[e]{e}< \sqrt[e]{\pi}$. After some work I was able to figure out the order. How can one ...
12
votes
7answers
1k views

How do we prove this logarithm?

Given: $$\dfrac{\log x}{b-c}=\dfrac{\log y}{c-a}=\dfrac{\log z}{a-b}$$ We have to show that : $$x^{b+c-a}\cdot y^{c+a-b}\cdot z^{a+b-c} = 1$$ I made three equations using cross multiplication : ...
4
votes
2answers
49 views

Find the solution to the differential equation

Assume $x\gt 0$ $$x(x+1)\frac{du}{dx} = u^2$$ $$u(1) = 4$$ I started off by doing some algebra to get: $$\frac{1}{u^2}du = \frac{1}{x^2+x}dx$$ I then took the partial fraction of the right side of ...
3
votes
4answers
183 views
+50

The value of the logarithmic expression can never be $\ldots$

The value of the logarithmic expression $\log_x \dfrac{x}{y} +\log_y \dfrac{y}{x},\text{where}\quad x\geq y>1\quad$ can never be $\bf\text{options}$ a.) $-1\quad$ b.)$\quad0.5\quad$ c.) ...
2
votes
2answers
35 views

Solving a problem involving $\log$ function

If $$a = \log_23 , b = \log_52$$ then what is $\log45$ ? (I have to define $\log45$ using $a$ and $b$) What I did : $$\log45 = 2\log3 + \log5$$ $$\log45 = \log2\left(2a + \frac1{b}\right)$$ Stuck ...
1
vote
2answers
121 views

Simplifying $\frac{\log(x)}{x}=y$.

I am trying to find the value of $r$ where the Rule of 72 will accurately estimate an investment's doubling time. Put simply, the Rule of 72 requires that 72 be divided by the interest percentage per ...
1
vote
2answers
39 views

Find when the population is growing the fastest, under the logistic model

The population $P$ of an island $y$ years after colonization is given by the function: $\displaystyle P = \frac{250}{1 + 4e^{-0.01y}}$. After how many years was the population growing the fastest? ...
0
votes
2answers
36 views

Raising/lowering with natural logs

I had a question on a test, and while I have already figured out that I should have done u substitution (I was running out of time and my brain froze), I was wondering if the following would be legal? ...
5
votes
1answer
142 views

Why are logarithms of trigonometric functions useful?

I have noticed that in many trigonometric tables the logarithm of the trigonometric values are given. Why this is given and not the actual values of the trigonometric functions? For example, instead ...
0
votes
2answers
917 views

Find Log equation from data points

I have the following data points, (left hand column goes from 0-127, right hand column goes from 30-22000 hz. Is there any calculator I can use to find a "log" function of this data, so that it comes ...
0
votes
1answer
29 views

Logarithmic system of equations

Solve these equations $$\log x+\frac{\log x+8\log y}{\log ^2x+\log^2y}=2$$ $$\log y+\frac{8\log x-\log y}{\log^2x+\log^2y}=0$$ Does an elegant solution exist? If not, how do I solve two cubic ...
0
votes
1answer
32 views

applying logarithm law question

Here is my equation (below) on which I am applying log $X=\frac{a}{b}\left ( c-d \right )$ so far I applied it as $\log X=\log(a)-\log(b)+\left [ \log\left ( c \right )-\log\left ( d \right ) ...
1
vote
1answer
41 views

Prove that $\log_a(1/x)=-\log(x)$.

I thought to write $$\log_a(1/x)=\log_a(x^{-1})=-\log_a(x)$$. But it has two problems: when x.0 and on the other problem it doesn't mention any condition. How should I solve it in each of them?
0
votes
1answer
38 views

Integral - complex exp. term

Does anyone know a suitable method to integrate and/or know the answer to: $\int\limits_{-\pi}^{\pi}$ $\log\Big[\tfrac{2 - a\exp({-it})}{1 - a\exp({-it})}\Big] $ ${\mathrm{d}t}$, for constant $|a|$ ...
0
votes
0answers
21 views

Using these number's logarithm with base 10 compare a with b when knowing $log_3(a)=log_5(b)$. [on hold]

Using these number's logarithm with base 10 compare a with b when knowing $$log_3(a)=log_5(b)$$. Please don't use this : log(a)/log(3)=log(b)/log(5)
2
votes
3answers
41 views

Solving an equaiton which includes $log$ as both base and exponent

Q: If $$9x = x^{\log_3x}$$ then what is $x$ ? I can't solve it. I have tried to use identities in my book but i think they are useless for this question. I need a hint
3
votes
2answers
51 views

Basic Logs, simplifying

So basically the question goes: $\log_{14} 2 = a$, $\log_{14} 3 = b$, solve for $\log_7 24$. I have attached my work so far and the answer is ${3a+b}\over{1-a}$... I just got stuck at the end. Thanks
0
votes
2answers
39 views

Is $x^x$ logarimithic? What about $x^{-x}$? And $-x^x$ and $-x^{-x}$? [on hold]

Is $x^x$ logarmithic? What about $x^{-x}$? And $-x^x$? Then what about $-x^{-x}$? Also, what is the summation function for Euler's number?
0
votes
0answers
32 views

Evaluating $ \int \cot^{2} \! \left( \frac{\pi}{\lfloor \log(x) / \log(3) \rfloor} \right) ~ \mathrm{d}{x} $.

How would I evaluate $$ \int \cot^{2} \! \left( \frac{\pi}{\lfloor \log(x) / \log(3) \rfloor} \right) ~ \mathrm{d}{x}? $$
1
vote
2answers
37 views

given that ${\log_9 p} = {\log_{12} q} = \log_{16}(p+q)$ find the value of $q/p$

This is not homework, it's just a brain teaser which I can't solve, just some hints should be sufficient, I know that from this we get: $$ (1/4)\log_2(p+q) = (1/2)\log_3 p = \frac{\log_3 q}{1+2\log_3 ...
0
votes
3answers
37 views

Solving an equation involving $\log_{10}$

If $$\log_{10}(x)\log_{10}(2) = 2$$ What is $x$ ? WolframAlpha says $x = e^{\frac2{\log_{10}(2)}}$ But i don't understand why it is.. Please explain it. Thanks
0
votes
2answers
47 views

Logarithm : Finding unknown log base? [closed]

Find $x$ : $$\log_4(8x^2)=\log_x(2)$$ I'm completely stuck on this question, can anyone help ?
1
vote
2answers
89 views

Taking an infinite number of logarithms

Let $n$ and $k$ be two integer parameters ($n\geq k$, if that matters). Define the following function: $\text{LOG }x=\max(\log{x},1)$ What is the limit of the following sequence as a function of $n$ ...
0
votes
1answer
35 views

Reversing equation with a logarithm and exponent

This is my equation: $$ x=y^{3.333+(-1(0.5\times \log_{10}(10-y)))} $$ It will solve for x, with input of any y. I want to solve for y with input of any x.
1
vote
4answers
45 views
2
votes
2answers
59 views

How to solve the derivative of $b^x$ using the defintion

I know that the derivative of $b^x$ is just $b^x \log{(b)}$, and I've seen it being derived using chain rule and such (not that I understand how it's done, I just learned about $e$ today so using the ...
10
votes
1answer
328 views

An equivalent for $\int_0^1\left(\frac{1}{\log x}+\frac{1}{1-x}\right)^n\;dx$

Set $$ I_n :=\int_0^1\left(\frac{1}{\log x} + \frac{1}{1-x}\right)^n \:\mathrm{d}x \qquad n=1,2,3,.... $$ We have $$I_1 =\gamma, \quad I_2 =\log (2 \pi) - \frac 32, \quad I_3 = 6 \log A - ...
-5
votes
2answers
43 views

What is e^e^x? Also, what is log e^e^x to the base e, i.e, ln(e^e^x)? [closed]

What is e^e^x? Also, what is log e^e^x to the base e, i.e ln(e^e^x)? Thank you.
0
votes
0answers
22 views

Baby-step Giant-Step algorithm to calculate value in new base

Using the Baby step–giant step algorithm I am trying to determine $log_{2}(7)$ in base $1$3. Let $p = 7$. Set $n$ to the least integer greater than $\sqrt p$: $n = 3$. So for baby step, I started off ...
0
votes
1answer
42 views

For which values of $a$ does this equation have a solution(s)?

The equation in question is $$\log_5x*(\log_5(2*\log_{10}a-x)*\log_x5+1)=2$$ Tried working this down with the rules of logarithms, got it down to a quadratic equation of $x$ with $a$ as one of its ...
0
votes
2answers
36 views

Log of many Logs

How can I compute the values of $n$ for which the following expression exists? $$\log_e(\log_e(\log_e(\log_e(\ldots\log_e(n))))$$ It is for instance apparent that when $n = e$, the second ...
1
vote
1answer
51 views

Integral of ln (3x) / x

I believe this should be a simple problem but I don't have an answer key to confirm if this is right, and some of the similar questions I can find online seem to be giving more complicated solutions. ...
-1
votes
3answers
56 views

Value of $x$ when $5 + \log x = \log \left(x^6\right)$

Find the value of $x$ when $$5 + \log x = \log \left(x^6\right)$$ I've tried many times to solve this, however I can't seem to find a correct (consistent) answer. My solutions range from $$x = e, x ...
1
vote
1answer
419 views

logarithmic function between two points

I need to find the logarithmic curve between two points $$A(0,5),\quad B(180,9)$$ We know that the formula for logarithmic function is: $\;f(x) = \log(x)\,\;$so $$ 5 = \log(0),\quad 9 = \log(180)$$ ...
1
vote
1answer
45 views

Simple Logarithmic question.

I was just wondering if i can do this. Q. Solve $\log_{9}24=x $ $\implies9^x =24$ $\implies3^{2x}=2^3 3$ $\implies\log_3(3^{2x})= \log_3(2^3 3)$ $\implies2x=2 (3)^{1/3}$ $\implies x=3^{1/3} $ ...
1
vote
3answers
45 views

How do I find the critical points of this function involving e?

I have the function: $$g(x)={{1 \over \sqrt{2 \pi}} \cdot e^{{-(x-2)^2}/2}}$$ Through very tedious differntion, I got to: $$g'(x) = {{{-(x+2)} \cdot {e^{{-(x-2)^2}/2}}} \over {2 \pi}}$$ Setting ...
0
votes
2answers
26 views

Converting log form of equation into linear form

I am trying to convert part of an equation from its log form into a linear form. Specifically, I am trying to convert $10^{4 log (x)}$, into $x^4$, but I'm really unsure of how to get from this first ...
1
vote
4answers
50 views

Prove that $\log_a(b)=\log(b)/\log(a)$

Prove that $$\log_a(b)=\log(b)/\log(a)$$ I don't know how to solve it, but I need to prove it so solve a problem.
0
votes
2answers
62 views

Prove that $\log_a(b)=-\log_b(a)$

Can you prove that: $$\log_a(b)=-\log_b(a)$$ I just thought that it should equal $$\frac{\log(b)}{\log(a)}.$$ but I don't think anything else.
0
votes
2answers
154 views

simple inverse function question

Functions in the form of $y = f(x)$ describe various sorts of line. In a quadratic line, for every extra unit in $x$, then $y$ increases by roughly $2x$. A line where for every extra unit in $x$, ...
1
vote
1answer
24 views

How do I simplify a Multivariable expression involving derivatives of logarithms?

I have this expression I got after a lot of calculation: $$\sigma =\frac{d\log\left(\frac{b(x,y,\rho)}{r(x,y,\rho)}\right)}{d\log\left(\frac{ 2 ...
5
votes
5answers
259 views

How to evaluate $\int_0^1 \frac{\ln(x+1)}{x^2+1} dx$

This problem appears at the end of Trig substitution section of Calculus by Larson. I tried using trig substitution but it was a bootless attempt $$\int_0^1 \frac{\ln(x+1)}{x^2+1} dx$$
0
votes
1answer
11 views

Logarithm Subject of Formula

$G_{dB}(f) = −10 \log_{10}(1 +\left(\frac f{f_3}\right)^2N)$. I will like to make $N$ the subject of the formula. Any lead on how to achieve this will be appreciated.