Questions related to real and complex logarithms.

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0
votes
1answer
27 views

How do I prove that ab+5(a-b)=1

If $\log_{12}18=a$ and $\log_{24}54=b$ then how do I prove $ab+ 5(a-b)=1$? I figured that out it's $\log_ab$ and $\log_{2a}3b$ but how do I solve it?
0
votes
2answers
29 views

How do i get from $x^{(\log(x))}=10000 $ to $\log(x)^2=\log(10000)$

I'm looking at the solution for a math problem I'm trying to solve and can't comprehend the following step: From: $$ x^{\log_{10}(x)}=10000 $$ To: $$ {\log_{10}(x)}^2=\log_{10}(10000) $$ Is there a ...
0
votes
2answers
20 views

Right angled triangle log

If $a,b$ and $c$($c$ is the hypotenuse) are sides of a right triangle then prove $$(\log_{c+b}a)+(\log_{c-b}a)=2(\log_{c+b} a )\cdot(\log_{c-b}a)$$ The bases are different so can't quite figure out ...
1
vote
4answers
39 views

Domain of the function $f(x) = \sqrt{\frac{3^x-4^x}{x^2-4x-4}}$ will be?

I tried solving this question by $1.$ $-1$ and $4$ will not be in domain because denominator can not be zero . $2.$ Either both denominator and numerator will be positive or negative so that whole ...
3
votes
2answers
31 views

Way to calculate exponent in congruent equation

I want to solve $$ 5^{x} \equiv 21 \pmod {23} $$ Is there a way to get the $x$ without trial & error?
1
vote
0answers
13 views

How do we get $S(m) = S(m/2) + \lg m$ from $T(n) = T(\sqrt{n}) + \lg\lg n$?

I am confused about example we got today in class. Here is a recurrence and I am not sure how we got $S(m)=S(m/2)+(\lg m)$ $$T(n)=T(\sqrt{n}) + (\lg\lg n) $$ Let $$m =\lg n$$ $$S(m)=S(m/2)+(\lg m) ...
0
votes
0answers
11 views

Logarithm with logarithm table

Out of curiosity, I wanted to use a log table to find the the logarithm of 347,5. (I always used a calculator) The first three digits (347) give me 5403 for the mantissa and for the fourth figure (5) ...
1
vote
1answer
26 views

A problem with logarithms

If $\log(a+b+c)=\log(a) + \log(b) + \log(c)$, prove that $$\log\left(\frac{2a}{1-a^2} +\frac{2b}{1-b^2} +\frac{2c}{1-c^2}\right) = \log\left(\frac{2a}{1-a^2}\right)+ ...
1
vote
1answer
20 views

Finding out the logarithmic function for the situation below

The situation reads as follows: There are 3000 barbs in a pond and every year 20% of the barbs die and then 1000 new barbs come to the pond. A logarithmic function needs to be plotted to graph ...
0
votes
3answers
44 views

How does $\log_2(A)-\log_2(B)+\log_2(c)$ not equal $\log_2(\frac{Bc}{A})$

$\log_2(A)-\log_2(B)+\log_2(c)$ How does this equal $\log_2(\frac{Ac}{B})$? Does it not follow from the order of operations that it would be addition first then subtraction? I'm having hard time ...
1
vote
4answers
47 views

A rational number which is 50 times its own logarithm to the base 10 is?

This question is from Advanced problems in mathematics for jee . I got it as a challenging question. I tried it in this way 50 log x base 10 = x But there seemed no solution for it as per my ...
0
votes
0answers
15 views

Proof that specific integral has no analytical solution.

The following integral \begin{align} \int_{0}^{1} dy \: \frac{\ln\left[\frac{\sigma_{12} y + \sigma_{13} (1-y)}{\sigma_{23}(1-y)y}\right]}{-\sigma_{23} y(1-y)+\sigma_{12} y +\sigma_{13} (1-y)} ...
2
votes
4answers
251 views

Explain this inequality, related to logarithms

I am trying to understand a proof of Stirling's formula. One part of the proof states that, 'Since the log function is increasing on the interval $(0,\infty)$, we get $$\int_{n-1}^{n} \log(x) dx ...
-1
votes
2answers
121 views

Exponential and logarithmic functions [on hold]

The temperature, $T (C^◦)$, of a quenched steel plate at time t is given by $$\large{T = f(t) = 10 + \frac{85}{1 + e^\frac{\large{\textbf{t}}−60}{10}}}$$ where $0 ≤ t ≤ 100$ measured in seconds. (a) ...
5
votes
5answers
88 views

Is it possible to prove this? $\ln(\frac{x}{x-1}) < \frac{100}{x} $ for $ x > 1$

$-\ln(1-(\frac{1}{x})) < \frac{100}{x} $ for $ x > 1$ is what I want to prove. I pulled a negative sign out and I got $\ln(\frac{x}{(x-1)}) < \frac{100}{x} $ for $ x > 1$. How do I ...
0
votes
2answers
33 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
2answers
29 views

Which is the limit of the following sequence? [on hold]

Calculate the following limit without using L'Hopital rules or integrals: $$\lim_{n\to ∞}\sum_{k=2}^n \frac{1}{k\ln k}$$
-2
votes
1answer
42 views

Simplifying logarithms into a single log using log Laws [on hold]

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 ...
0
votes
2answers
29 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 ...
1
vote
1answer
34 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 ...
2
votes
1answer
42 views

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

Is this mathematically legit? $$\lim_{x\to\infty}\ln(f(x))=\ln(\lim_{x\to\infty}(f(x))$$
-5
votes
0answers
24 views

Compare a with b knowing that $log_3{a}=log_5{b}$. [closed]

You should solve it by turning these logarithms into $\log$ with base $10$. Please don't use $\frac {\log a}{\log b}=\frac {\log b}{\log 5}$
1
vote
2answers
37 views

Solve the exponential equation, by using $\log_{10}$ of of the sides of it. [closed]

Solve the exponential equation, by using $\log_{10}$ of of the sides of it. The equation is: $5^{x-1}=2$
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
20 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
35 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 ...
-1
votes
1answer
24 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
35 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 $$ ...
0
votes
1answer
31 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
2answers
28 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
38 views

Multiplying logarithms of different bases [closed]

How do you multiply the following logs... $$\log_5(n) * \log_2(n)$$
1
vote
1answer
19 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 ...
1
vote
1answer
21 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 ...
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 : ...
2
votes
2answers
38 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
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 ...
1
vote
2answers
52 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
47 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
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 ...
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 ...
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
39 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
42 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
0
votes
2answers
51 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 ?
0
votes
1answer
36 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.