Questions related to real and complex logarithms.

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0answers
4 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 ...
8
votes
7answers
224 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 : ...
2
votes
2answers
33 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 ...
4
votes
2answers
44 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 ...
1
vote
2answers
33 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? ...
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 ...
0
votes
1answer
27 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 ...
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
38 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
36 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? [on hold]

Find $x$ : $$\log_4(8x^2)=\log_x(2)$$ I'm completely stuck on this question, can anyone help ?
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.
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 ...
1
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4answers
45 views

Logarithm Question (Find x) [on hold]

How to solve x for $$x^{2\log_{10}x}=\frac{x^5}{100}$$?
-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
54 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
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
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 $n$ ...
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
49 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
61 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.
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 ...
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.
4
votes
1answer
87 views

If $\frac{x-1}{e^x-1} = y$ then $x=?$

I have following equation: $$\frac{x-1}{e^x-1} = y$$ I want to solve this equation such that I have the value of $x$ in the term of $y.$ i.e. inverse of the equation
1
vote
2answers
18 views

Limit log-sum of exponentials

I'm trying to compute the following limit: $$\lim_{\lambda \rightarrow \infty} \frac{1}{\lambda}\log\sum_{i=1}^n \exp[\lambda a_i]$$ I tried to solve with L'hoptials: $$= \lim_{\lambda \rightarrow ...
0
votes
1answer
30 views

Transpose exponential equation [closed]

Could somebody please help with transposing the following equation to isolate x to the left side of the equation to solve for x? $$ y = 10^{1.830 \log(x)} + 2.686 $$
0
votes
1answer
35 views

Using log to take derivative of a function

Is it safe to say that if $\frac{d}{dx}ln(f)= g $ for some functions f and g, then $\frac{d}{dx}f = e^{g}$? Why or why not? (novice high schooler here)
1
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1answer
34 views

Combining log terms

I have this particular problem. We have to combine the log terms into a single log term: $$\dfrac{(2\ln a- \ln b - 5\ln c)}{2}$$ I did it in the following way : $$''~= \ln a -\frac{1}{2}\ln b - ...
1
vote
4answers
247 views

Natural log limit question

I have to find $$\lim_{n\to\infty}\left(\ln(n-1)-\ln(n)\right)$$ I'm pretty sure I need to solve this using the asymptotes. So if I use the rule for logs I can do lim (ln((n-1)/n)) and I know that ...
1
vote
1answer
33 views

Minimum value of function $f(x)=x+\log_2(2^{x+2}-5+2^{-x+2})$ out of 5 options

Minimum value of function $f(x)=x+\log_2(2^{x+2}-5+2^{-x+2})$ out of 5 options A : $\log_2(1/2)$ B : $\log_2(41/16)$ C : $39/16$ D : $\log_2(4.5)$ E : $\log_2(39/16)$ I just... don't know how to ...
1
vote
0answers
34 views

Is this chain of inequalities correct?

Is this chain of inequalities correct? If not how to make it works? $$\frac{\ln \left( 1+x^3+y^3 \right)}{\sqrt{x^2+y^2}} \le \frac{\left( x^3+y^3 \right)}{\sqrt{x^2+y^2}} \le \frac{ \left( ...
3
votes
3answers
65 views

How to integrate $x\ln(x+1)$?

I am trying to compute $\int x\ln (x+1)\, dx$. I tried integrating by parts and ended up with: $$\int x\ln(x+1)\,dx = \frac{1}{2}x^2\ln(x+1) - \frac{1}{2}\int\frac{x^2}{x+1}\,dx$$ but I'm stuck here.
1
vote
2answers
53 views

Is $\log_{\cos x}(1)$ defined at $x=0+2k\pi$? [duplicate]

I have an equation like this: $\cos(x) ^ {\sin(x)} = 1$ I thought I would solve it like this: $\cos(x) ^ {\sin(x)} = 1$ $\sin(x) = \log_{\cos(x)}(1)$ $\sin(x) = 0 $ $x = 0+k\pi$ But I'm ...
1
vote
1answer
23 views

Does the following series of transformations of inequalities holds?

I am to calculate limit of the function $f(x,y)$ i am trying to apply squeeze theorem. Is the following series of transformations of this inequality correct? If not how to do this correctly? i.e. are ...
1
vote
1answer
29 views

Logarithm multivariable limit $\frac{\ln(1+x^3+y^3)}{\sqrt{x^2+y^2}}$

Find multivariable limit $$\lim_{\left( x,y \right) \rightarrow (0,0)}\frac{\ln(1+x^3+y^3)}{\sqrt{x^2+y^2}}$$ I was trying to find and inequality i've found out that: ...
1
vote
1answer
34 views

Finding the time for an epidemic/computer virus to infect a population

Question: "Suppose a computer worm makes 2 copies of itself on another computer in one millisecond. Estimate the time that is needed to spread to a population of 1,000,000 computers" How would I ...
1
vote
0answers
23 views

1st order ODE separable

everyone! :-) I've a ODE question with I can't solve. It's here: ${dy\over dx} = {{xy + 2y-x-2}\over {xy-3y+x-3}} $ I tried the following: ${dy\over dx} = {{xy + 2y-x-2}\over ...
0
votes
3answers
36 views

How is this logarithmic identical transformation true? [closed]

$$x^{1-\log x}=1\Leftrightarrow(1-\log x)\log x=\log 1?$$ I don't know how it can be true.