Questions related to real and complex logarithms.

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0
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2answers
34 views

Meaning of exponent in logarithm?

I have this particular difficulty : $$\log_b^a(c)=x$$ I know it is different from power of base $\log_{b^a}(c)=x$, but what does it actually mean? The actual question that i got in paper was Find ...
0
votes
3answers
299 views

Why I can't calculate $0*log(0)$ but can $log(0^0)$

I got this doubt after some difficult in programming. In a part of code, i had to calculate: $$ x = 0 * Log(0) \\ x = 0*-Inf $$ and got $x = NaN$ (in R and Matlab). So I changed my computations to ...
0
votes
2answers
31 views

Simple math Question concerning the natural logarithm of Complex Number

There is this simple exercise, in which the complex number is given in polar form as z= mod=|10|,arg=322.75 degrees and i must find the ln of it. So to do that i must first convert the complex number ...
-2
votes
1answer
43 views

Simplifying logarithms into a single log using log Laws [closed]

I need some help on how to solve this question. $$1.5(\log_bx+2\log_by^4)-0.5(\log_b\sqrt x+\log_by^{1/3})$$ Shall appreciate some help on this. My Work: if possible, could u tell me if my ...
3
votes
5answers
254 views

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.) ...
1
vote
1answer
36 views

Unable to solve logarithm question

Given $$\dfrac{a(b+c-a)}{\log a}=\dfrac{b(c+a-b)}{\log b}=\dfrac{c(a+b-c)}{\log c}$$ To prove: $$a^bb^a=b^cc^b=c^aa^c$$ What i tried is $$\log (a^z)=a(b+c-a)$$ and similarly for other two. I am ...
6
votes
3answers
117 views

How prove this $H_{2n}-H_{n}+\frac{1}{4n}>\ln{2}$

Show that, for every positive integer $n$, $$\dfrac{1}{n+1}+\dfrac{1}{n+2}+\cdots+\dfrac{1}{2n}+\dfrac{1}{4n}>\ln{2}$$ I know this ...
0
votes
2answers
49 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? ...
2
votes
1answer
43 views

Is this $\lim \ln(f(x))=\ln(\lim(f(x))$ valid?

Is this mathematically legit? $$\lim_{x\to\infty}\ln(f(x))=\ln(\lim_{x\to\infty}(f(x))$$
0
votes
3answers
32 views

Logarithm formula proof

Prove: $$x^{\log(y)}=y^{\log(x)}$$ I have been trying this for the past 1 hour, still cant prove it. I started with $$\log_b(y)=m$$ $$\log_b(x)=n$$ To show: $$x^m = y^n$$ How do i proceed? :
2
votes
1answer
21 views

Find curve that fits (min, mean, max) to (0, 0.5, 1) [closed]

I'm trying to use the fact that $log(1) = 0$ and $log(\sqrt{e}) = 0.5$ and $log(e) = 1$ to write a map from a set of data points to a value between $0$ and $1$ such that: $f(min) = 0$, $f(mean) = ...
0
votes
2answers
36 views

Showing that for continuous logarithms $g_1, g_2$ of a function on a connected set, the difference $g_1 −g_2$ is a constant

If $S$ is connected, $\ f$ is continuous and has continuous logarithms $g_1$ and $g_2$ on $S$, and continuous arguments $\theta_1$ and $\theta_2$, then $g_1 −g_2$ and $\theta_1-\theta_2$ are ...
0
votes
1answer
36 views

How to bound this difference between two logarithmic expression

I want to bound the difference between two logarithmic expression shown below with a constant number i.e not function of $x,y,z$ where $x,y,z \in \mathbb{C}$. The difference is $$ ...
-1
votes
1answer
25 views

Solving a logarithmic equation with variables on each side

Okay, so while doing a problem for my calculus class I was required to graph two functions in order to see where they intersect, as according to my teacher there is no way to solve it analytically. ...
0
votes
1answer
33 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
24 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
314 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
34 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
46 views
1
vote
1answer
22 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
209 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
54 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 ...
2
votes
2answers
44 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
124 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
65 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? ...
5
votes
1answer
146 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
1k 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
31 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
34 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
42 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
44 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 $|a|$ ...
2
votes
3answers
43 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
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
39 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
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
37 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
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4answers
45 views
2
votes
2answers
60 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
332 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 - ...
0
votes
0answers
26 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
43 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
38 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
53 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
57 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
469 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
48 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} $ ...