Questions related to real and complex logarithms.

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0
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1answer
13 views

What is the difference in exponential of log and ln

What is the difference in exponential of log and ln? For example, exp(ln(sqrt(2)) and exp(log(sqrt(2)) What will be the answer ...
4
votes
5answers
46 views

Limit of $2^{(\log_2 n)^2}/2^{(\log_2 n)^3}$

I am trying to find the following limit $$\lim_{n \to \infty} \frac{2^{(\log_2 n)^2}} {2^{(\log_2 n)^3}}$$ I really don't know where to start and any help would be appreciated!
0
votes
2answers
36 views

A quick Logarithm question

Did I do this problem right? $\log y= 10 + 2(\log x)$ $y= 10^{10} + x^2$? Edit: How does $10^{\log (x^2)}= x^2$? I can't seem to find an explanation for this. All the websites that I found already ...
1
vote
2answers
48 views

How to evaluate $45^\frac {1-a-b}{2-2a}$ where $90^a=2$ and $90^b=5$ without using logarithm?

Let $90^a=2$ and $90^b=5$, Evaluate $45^\frac {1-a-b}{2-2a}$ I know that the answer is 3 when I used logarithm, but I need to show to a student how to evaluate this without involving logarithm. ...
3
votes
1answer
49 views

Prove $x^n < n^n 2^x$

Given that $$x < 2^x$$ is always true, use it to prove that $$x^n < n^n2^x$$ Here are the steps that I've taken so far: Reduce $$x < 2^x$$ to $$\log(x) < x$$ Then $$x^n < n^n2^x$$ ...
2
votes
5answers
71 views

Solve logarithmic equation $ 3^{\log_3^2x} + x^{\log_3x}=162$

Find $x$ from logarithmic equation $$ 3^{\log_3^2x} + x^{\log_3x}=162$$ I tried solving this, with basic logarithmic laws, changing base, etc., but with no result, then I went to wolframalpha and it ...
1
vote
2answers
33 views

Conversion from sum of product to product of sum

I do not understand how did he convert from this to this. Source :http://cs229.stanford.edu/notes/cs229-notes1.pdf Page 18
8
votes
1answer
81 views

What is $\lim_{n\to\infty}2^n\sqrt{2-\sqrt{2+\sqrt{2+\dots+\sqrt{p}}}}$ for $negative$ and other $p$?

This was inspired by similar posts like this one. Define the function, $$F(p) = \lim_{n\to\infty}2^n\sqrt{2-\underbrace{\sqrt{2+\sqrt{2+\dots+\sqrt{p}}}}_{n \textrm{ square roots}}}$$ We know that, ...
1
vote
1answer
66 views

$(1)$ $a\boxplus b=a+b-ab$ for all $a,b \in F$. $(2)$ $a\boxdot b=1-t^{\log_t{(1-a)}\log_t{(1-b)}}$ for all $a,b\in F$

Let $1<t\in \mathbb{R}$ and let $F=\{a\in \mathbb{R}: a<1\}$. Define $\boxplus$ and $\boxdot$ on $F$ as follows: $a \boxplus b=a+b-ab$ for all $a,b \in F$. $a \boxdot b=1-t^{\log_t ...
2
votes
1answer
48 views

Simplified is 0? $\log_{3} 9x^4 - log_{3}(3x)^2 $

I have tried to solve this multiple ways, but I keep getting $2\log_{3}x$. According to the answer key, it is supposed to work out to 0, but I'm not seeing it. Can someone point me in the right ...
2
votes
4answers
49 views

Prove $\log(x) < n(x)^{1/n}$, for all positive integer values of $n$, and $x > 0$

Given that $$lg(u) < u$$ is always true, how do we use that to prove that $$lg(x) < n(x)^\frac 1n$$ These are the steps that I have taken so far: $$1: lg(x) < n(x)^\frac 1n$$ $$2: \frac ...
1
vote
1answer
29 views

Computing the limit of $(\log n)^{0.5}/\log n^{0.5}$

$$\lim\limits_{n\to\infty}\frac{(\log n)^{0.5}}{\log n^{0.5}}$$ I'm really not sure where to begin with this. Are there some basic laws of logs that I should apply first?
1
vote
4answers
35 views

Asymptotic behaviour of the logarithm

In this post, the poster suspected that the $\log$ function would eventually flatten out and approach a straight line. We all know this isn't true of course. But then a commenter pointed out this: ...
1
vote
3answers
36 views

Prove that from the equalities, $\frac{x(y+z-x)}{\log x}=\frac{y(x+z-y)}{\log y}=\frac{z(y+x-z)}{\log z}$ follows $x^yy^x=y^zz^y=z^xx^z$.

Problem : Prove that from the equalities, $$\frac{x(y+z-x)}{\log x}=\frac{y(x+z-y)}{\log y}=\frac{z(y+x-z)}{\log z}$$ follows $$x^yy^x=y^zz^y=z^xx^z$$. My approach : $$\frac{x(y+z-x)}{\log ...
7
votes
3answers
160 views

A reason for the value of $\int_{0}^{1}\log{(x)}\log{(1-x)}\,\mathrm{d}x$

In this .pdf document, which is just a list of Putnam-style undergraduate-level problems from various sources, the third question is as I have stated it below (up to a change of notation). ...
0
votes
3answers
46 views

Simplifying two logarithms with different bases

I am being asked to simplify: $(\log_4 7)(\log_7 5)$ How can this be simplified given that the bases are different?
0
votes
1answer
34 views

Logarithm of the dot product of two vectors.

Let $\vec{u}$ and $\vec{v}$ be two vectors and $\vec{u}\cdot\vec{v}$ be their dot product. My question is that how to take the logarithm of the dot product, that is, how do we find ...
0
votes
1answer
44 views

Partial integration of $\sin x\log(y-1)$ w.r.t. $x$

If I have the function $\sin x\log(y-1)$ and I want to partially integrate it w.r.t. $x$ then what happens to $\log $? Would the solution be: $-\cos x \log(y-1)$ and how? Isn't $\log(y-1)$ a function ...
0
votes
3answers
40 views

$a^{\log_g b} = b^{\log_g a}$?

$g^{\log_g a} = a$, because it equals $a^{\log_g g}$. Does this mean that $a^{\log_g b} = b^{\log_g a}$? Note: thanks whoever edited it to proper markup
0
votes
0answers
43 views

tricky derivative with logarithm of sum

I'm having trouble understanding the solution of a limit. It involves a formula for measuring certainty of a discrete probability distribution. Given a set of values $p_j$ which sum up to 1, find the ...
-3
votes
0answers
17 views

what is formula to this eqution [(256)16]1/32+[(169)6]1/12 [closed]

how to solve this equation [(256)16]1/32+[(169)6]1/12 what is formula of this? What is the closed form expression for this? What is the right domain for this Hamiltonian 2? what is the right ...
0
votes
0answers
9 views

Tweaking function to reduce the rate of decay of a logarithmic based curve

Im not even sure if this is possible or perhaps I may need to use a different function altogether but I currently have one that looks like this: $$y = a\log(x+b)+c$$ That produces the red curve ...
2
votes
1answer
36 views

Name for some kind of logarithmic norm/error

As known $(\mathbb R, +)$ and $(\mathbb R^{+}, \cdot)$ are isomorphic with $\exp:\mathbb R\to\mathbb R^{+}$ as an isomorphism. When I transfer the absolute value $|\cdot|$ on $(\mathbb R, +)$ via ...
-5
votes
1answer
35 views

Hard logarithm question [closed]

$\log_a x$, $\log_b x$ and $\log_c x$ are in ap, where $x \ne 1$, then show that $$c^2=(ac)^{log_a b}.$$
3
votes
2answers
80 views

evaluating some limits with $\ln(x)$

I don't understand how to prove these results. $\lim\limits_{x \to +\infty}\dfrac{\ln{x}}{x} = 0$ $\lim\limits_{x \to 0^{+}}x\ln{x} = 0$
0
votes
2answers
78 views

How many pairs $(m, n)$ exist?

For certain pairs $ (m,n)$ of positive integers with $ m\ge n$ there are exactly $ 50$ distinct positive integers $ k$ such that $ |\log m - \log k| < \log n$. Find the sum of all possible ...
3
votes
1answer
25 views

Primitive of $dz/z$ is a branch of log

Let $D$ a connected open set of $\mathbb{C}$. A continuous function $f:D\to \mathbb{C}$ is a branch of log if $e^{f(t)}=t$ on $D$. In my book (Cartan) it is written that if $F$ is a primitive of the ...
14
votes
3answers
188 views

Prove that $\int_0^1 \frac{1}{1+\ln^2 x}\,dx = \int_1^\infty \frac{\sin(x-1)}{x}\,dx $

I've found the following identity. $$\int_0^1 \frac{1}{1+\ln^2 x}\,dx = \int_1^\infty \frac{\sin(x-1)}{x}\,dx $$ I could verify it by using CAS, and calculate the integrals in term of ...
1
vote
1answer
83 views

Solve $\log_9 (a) + \log_{12} (b) = \log_{16} (a+b)$ for $a/b$

The question: $$\log_9 (a) + \log_{12} (b) = \log_{16} (a+b)$$ solve for $a/b$. It gives hints: put it all in terms of x. $$9^x=a$$ $$12^x=b$$ $$16^x=a+b$$ Now prove that: $b^2=a(a+b)$ I did and ...
0
votes
1answer
29 views

A question to do with logarithms?

$\log_3x^3 + {3\over \log_3x} =4$ Ok, the way the computer has put it makes it look weird.But it is :log to base 3 of "x" to power 3 plus 3 divided by log to base 3 of x is equals to "4". This ...
3
votes
3answers
48 views

Solving logarithmic equations

The equation that I'm trying to solve is: $$\log _{5x+9}(x^2+6x+9)+\log _{x+3}(5x^2+24x+27)=4$$ Using algebra and principles of logarithms I managed to get the equation down to $$\frac{2\left(\log ...
0
votes
0answers
16 views

Help finding a formula to fit the data - both axes are logarithmic?

I'm an electrical engineer trying to come up with a formula to turn a measurement of light level into lux. The photoresistor changes resistance logarithmically. Lux is also logarithmic. So the data I ...
4
votes
4answers
78 views

Is $\ln(x^{p(x)}) = p(x) \ln(x)$?

I am trying to prove that: $x^{\frac{\ln(\ln(x))}{\ln(x)}} = \ln(x)$ My "solution": $e^{\ln\left(x^{\frac{\ln(\ln(x))}{\ln(x)}}\right)} = e^{\frac{\ln(\ln(x))}{\ln(x)} \ln(x)} = e^{\ln(\ln(x))} = ...
4
votes
1answer
63 views

Is there another function with a property like the log?

Is there another differentiable monotone increasing (or decreasing) function $ f:\mathbb{R} \rightarrow \mathbb{R} $ with a property that $ f(xy) = f(x) + f(y) $, like the log-function has it?
2
votes
6answers
38 views

Solving for $x$ in an exponential equation

Say we the following equation $$F(x) = \frac{\exp(a+bx)}{1 + \exp(a+bx)}$$ Now we set $x=0$ and we want to solve for $a$ as a function of $F_0$. So that, we have: $$F_0 = \frac{\exp(a)}{1 + ...
2
votes
1answer
49 views

Logarithm question, does $\ln \sqrt{7}$ equal to zero?

I got this from my workbook solution, was able to solve the question for the most part but stuck in the last sentence. $$7(\ln\left|x+\sqrt{x^2-7}\right|-\ln\sqrt{7}) + c = 7 ...
0
votes
2answers
29 views

Show using logarithms that the first equation can be transformed into the second.

Show using logarithms that if $y^k = (1-k)zx^k(a)^{-1}$ then $y = (1-k)^{(1/k)}z^{(1/k)}x(a)^{(-1/k)}$.
9
votes
5answers
547 views

How to solve an exponential and logarithmic system of equations?

$$ \left\{\begin{array}{c} e^{2x} + e^y = 800 \\ 3\ln(x) + \ln(y) = 5 \end{array}\right.$$ I understand how to solve system of equations, logarithmic rules, and the fact that $\ln(e^x) = e^{\ln(x)} ...
1
vote
2answers
118 views

Solving $x^{2n} = \frac{1}{2^n}$ for $x$

What is the principle behind solving for a variable that is raised to another variable? I came across this problem doing infinite sums: I had to solve the equation $$x^{2n} = \frac{1}{2^n}$$ for ...
1
vote
3answers
40 views

How to find range of a logarithmic function?

How do I find the range of these logarithmic functions? \begin{align} & \ln(3x^2 -4x +5), \\ & \log_3(5+4x-x^2). \end{align} how should I approach questions like this ? What I did: I found ...
1
vote
3answers
41 views

Proving: $\frac{2x}{2+x}\le \ln (1+x)\le \frac{x}{2}\frac{2+x}{1+x}, \quad x>0.$

$$\begin{equation}\frac{2x}{2+x}\le \ln (1+x)\le \frac{x}{2}\frac{2+x}{1+x}, \quad x>0\end{equation}$$ I found this inequality in this paper: http://ajmaa.org/RGMIA/papers/v7n2/pade.pdf (Equation ...
-1
votes
1answer
33 views

Natural Logarithms (Help) [closed]

Can I have an answer with working out to this question please, I am puzzled at the result I got and don't believe it is right. The question asks to 'Simplify by expressing as a single natural ...
0
votes
0answers
21 views

Manipulating product of two matrices

In a published paper I saw the following $$\log \left(\mathbf{I} + \mathbf{T}\mathbf {Hpp^HH^H}\right)= \log(1+\mathbf {p^HH^HTHp})$$ where uppercase means a matrix while lower case means vector ...
2
votes
1answer
68 views

Is $f(\sqrt{x}) = \sqrt{f(x)}$ true, where $f(xy) = f(x) + f(y)$ for all positives $x, y \in \mathbb{R}$?

Is $f(\sqrt{x}) = \sqrt{f(x)}$ true, where $f(xy) = f(x) + f(y)$ for all positives $x, y \in \mathbb{R}$? It's obviously false. But the point is that "can it be proved without using the fact that the ...
0
votes
1answer
34 views

When does function $(\log_b(x))^p$ change its curvature?

Consider $(\log_b(x))^p$ where $b$ is a constant $>1$; $x, p \in \mathbb R_+$. As we increase the value of $p$ (starting from 1), at specific value of $p$, the curve changes its shape from ...
1
vote
1answer
34 views

Taking Log to find MAXIMIZE summation of variables

I have been reading IEEE papers on communication and in several papers the authors formed objective function like: $\text{Maximize } \sum_k \log r_k $ to maximize the total rate of the system of ...
0
votes
0answers
19 views

What's the non-log form of this equation?

I found this equation in a book and I'd like to know what it would look like in 'non-log' form: $(a \log x + b \log N)^2 + 2g \log x + 2f \log N + C = 0$ Thanks.
1
vote
4answers
56 views

Integrating for a solution in terms of an natural logarithm

Evaluating the following integral: $$\int_1^2 \frac2{1-3x}\ dx$$ why do you have to take the factor of $-2/3$ out when evaluating the integral?
1
vote
0answers
42 views

solve $a\cdot e^{b\cdot x}+c\cdot ln(x)=0$

Is it possible to find the analytical solution of $a\cdot e^{b\cdot x}+c\cdot ln(x)=0$? Is that a transcendental equation?
2
votes
6answers
87 views

How do I solve for $t$ in this equation?

I know I'm supposed to use $\ln()$ to work it out, but I can't remember how it's done. Can anyone help? The equation is $$ 40e^{-t/5}=20 $$