Questions related to real and complex logarithms.
3
votes
2answers
37 views
Stuck on an 'advanced logarithm problem': $2 \log_2 x - \log_2 (x - \tfrac1 2) = \log_3 3$
I'm stuck on solving what my textbook calls an "advanced logarithm problem". Basically, it's a logarithmic equation with logarithms of different bases on either side. My exercise looks like this:
$$2 ...
1
vote
0answers
20 views
Generalised logaritmic function
I was wondering if there was a function that extends the domain of the following function to non-negative real numbers. For non-negative integer $n$ and real $y$, $y = f(x,n)$ is given by:
$$f(x,n) = ...
1
vote
1answer
45 views
How do I solve $\; 3^{2x+1}-10\cdot 3^x+3=0 \quad?$
Solve the following equation for $x$ : $ \quad3^{2x+1}-10\cdot 3^x+3=0 $
I am baffled to solve this equation. With graphing I have found the answers to be x=1 and x=-1. I would like to know how ...
2
votes
2answers
45 views
How to solve equations with logarithms, like this: $ ax + b\log(x) + c=0$
I encountered an equation of type $$ ax + b\log(x) + c=0$$ Here a, b, and c are constants. Does anyone know how to solve these type of equations? I guess this way:
$$\log(x)= \frac{c-ax}{b}$$
$$x= ...
0
votes
3answers
88 views
Upper Bound of Logarithm
For $1\leq x < \infty$, we know $\ln x$ can be bounded as following:
$\ln x \leq \frac{x-1}{\sqrt{x}}$.
Then what is the upper bound of $\ln x$ for following condition?
$2\leq x <\infty$
4
votes
1answer
72 views
Estimating $\sum_{p_2 \leq x} (\log p_2)^2$
This was an exercise to use the approach here to estimate the sum $\sum_{p_2 \leq x} \log (p_2)^2,$ in which $p_2$ are numbers containing two prime factors (repetitions allowed). $\pi_2(x)$ is the ...
1
vote
2answers
64 views
Filling in 'x' in a log function
if $3^5=x$ (exponential equation) converts to log form gives $log_3x=5$
that makes sense.
$$
3^5 = 243 \Rightarrow x=243
$$
So if I take the log form again: $log_3x=5$ and replace $x$ with $243$. I ...
2
votes
4answers
65 views
Is it possible to solve logarithm when base is unknown?
Is it possible to solve logarithm equation when the base of the logarithm is unknown but the result is known. Here is an example:
$$ \log_{X} (\frac{223}{150}) = 20 $$
This basically means that if x ...
0
votes
3answers
52 views
Ln Series Summation
I have been given:
$$\ln{n}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\text{ for sufficiently large }n$$
Which I can equate to $\ln(n)=\sum\limits_{i=1}^n \frac{1}{i}$
The series I need ...
1
vote
3answers
49 views
Evaluating $\lim\limits_{x\to 0^{+}} \frac{x}{\ln^2 x}$
How can I find:
$$\lim_{x\to 0^+} \frac{x}{\ln^2 x} $$
I know that the limit is $0$. I tried sandwich theorem but I don't know what could be bigger.
Thanks in advance.
2
votes
3answers
80 views
Can we *ever* use certain log/exp identities in the complex case?
This article on Wikipedia points out that certain identities for the log and exponential functions which are familiar from the real case require care when used in the complex case. Failures in the ...
1
vote
2answers
55 views
Finding Big-O with Fractions
I'd want to know how I can find the lowest integer n such that f(x) is big-O($x^n$) for
a) $f(x) = \frac {x^4 + x^2 + 1}{x^3 + 1}$
I've fooled around with this a bit and tried going from
$\frac ...
6
votes
1answer
118 views
+200
The positive root of the transcendental equation $\ln x-\sqrt{x-1}+1=0$
I numerically solved the transcendental equation
$$\ln x-\sqrt{x-1}+1=0$$
and obtained an approximate value of its positive real root $$x \approx 14.498719188878466465738532142574796767250306535...$$
...
4
votes
2answers
73 views
What is an effective and practical means to teach about natural logarithms and log laws to high school students?
My students are quite practically minded, and I have found that teaching them concepts in a practical manner to be very helpful (maths 'experiments'; modelling on the smartboard etc).
I am looking ...
1
vote
3answers
60 views
What does ($\ln x$) or ($\log x$) mean?
How does a logarithm followed by a variable read such as ($\ln x$) or ($\log x$). Is it $\log$ times $x$ or the $\log$ of $x$? I'm a little confused by this...?
0
votes
1answer
31 views
What role does 1/α play in the last integal?
Link of the page:http://imgur.com/N4uUeA9
α is defined as follows:http://imgur.com/6ESJUE8
Why is it here?I can't make any sense out of its use.
8
votes
4answers
565 views
Prove that $\log _5 7 < \sqrt 2.$
Prove that $\log _5 7 < \sqrt 2.$
Trial : Here $\log _5 7 < \sqrt 2 \implies 5^\sqrt 2 <7.$ But I don't know how to prove this. Please help.
2
votes
6answers
78 views
Help with differentiation of natural logarithm
Find $\;\dfrac{dy}{dx}\;$ given $y=\frac{\ln(8x)}{8x}$.
The answer is $\;\dfrac{1-\ln(8x)}{8x^2}\;$.
Can you show the process of how this is worked?
Thanks.
3
votes
9answers
161 views
Finding the definite integral $\int_0^1 \log x\,\mathrm dx$
$$\int_{0}^1 \log x \,\mathrm dx$$
How to solve this? I am having problems with the limit $0$ to $1$. Because $\log 0$ is undefined.
4
votes
2answers
68 views
How to Solve for Zero
$$4x^2e^{-x^2}-2e^{-x^2}=0$$
I took out a common factor of $2e^{-x^2}$ which got me to:
$2e^{-x^2}(2x^2-1)=0$
I'm not sure if taking out the common factor helped at all and I don't know where to go ...
3
votes
1answer
84 views
How is $(\arg F)(z)$ of complex function calculated?
My book on complex analysis introduces $\arg(z)$ and $\arg(z-a)$ after the complex logarithm is introduced. It shows that the two are just the oriented angles between $z$ and the point $z_0$ of the ...
0
votes
4answers
65 views
$a^{\log_b(c)} = c^{\log_b(a)}$ [duplicate]
I'm not sure how to start. My questions is how do you prove:
$$a^{\log_b(c)} = c^{\log_b(a)}$$ where $a,b,c > 0$.
0
votes
2answers
29 views
Exponential function passing through two points. [closed]
Find the values of $a$ and $b$ such that the exponential function $y=ab^x$ passes through the points $(1,36)$ and $(3,64)$.
2
votes
1answer
37 views
Derivative of $x\times n - 2^{\log_2 {x \times n}}$
I have a problem with solving derivative of $f(x)$ in this case:
$$f(x) = x\times 10^9 - 2^{\log_{10} x\times 10^9}$$
This is what I have:
$$f^\prime(x) = \lim_{m\to0} {f(x+m) - f(x)\over m}$$
$$= ...
-4
votes
1answer
62 views
What is derivative of$ x - 2^{\log n}$ [closed]
Wrong question. Needs to be deleted (already flagged...)
0
votes
1answer
94 views
Evaluation of the integral $\int \cos\omega t\ln\cos\omega t\,dt$
I am trying to evaluate an integral of the form
$$ \int \cos\left(\omega t\right) \ln \cos\left(\omega t\right) dt$$
and am unsure how to proceed.
I rewrote it as:
$$ \textrm{Re} \left\{\int dt ...
6
votes
4answers
126 views
Natural logarithms base $e$
Why is $e$ used as a base of natural logarithms everywhere?
Is the origin from the fact that exponential is the only function with the unique property of its differential and integral same and that ...
2
votes
1answer
37 views
Exponential practice exam question
Okay bear with me, this is one of those cumulative questions
The amount of a certain type of drug in the bloodstream t hours after it has been taken is given by the formula:
$$x = D{e^{ - {1 \over ...
0
votes
1answer
46 views
Does $i = -\frac{(2\;W({\pi\over2}))}{\pi}$
Let $x = -\frac{(2\;W({\pi\over2}))}{\pi}$, where $W$ denotes the Lambert W-function.
As
$${\log(i^2)\over i} = \pi$$
and $${\log(x^2)\over x}=\pi$$
Does $x = i$?
2
votes
3answers
58 views
Show that $x=2\ln(3x-2)$ can be written as $x=\frac{1}{3}(e^{x/2}+2)$
Show that $x=2\ln(3x-2)$ can be written as $x=\dfrac{1}{3}(e^{x/2}+2)$.
Is there a rule for this?
0
votes
0answers
18 views
Reduce Lethargy Equation
I need to prove that 1-((A-1)^2/2A)(ln((A+1)/(A-1)) approximately equals 2/(A+2/3)
I think that we can expand the ln to 2(1/A+1/(3A^3)+...) and so the first term multiplied by the polynomial and all ...
0
votes
3answers
67 views
$1500=P \times { (1 + 0.02) }^{ 24 }$, what is the value of $P$?
Hey guys could you please tell me what is the faster why to solve this equation. It's a compound interest equation and I'm stuck at the ${ (1 + 0.02) }^{ 24 }$ I really don't know how to proceed in ...
4
votes
5answers
106 views
How do I solve such logarithm
I understand that
$\log_b n = x \iff b^x = n$
But all examples I see is with values that I naturally know how to calculate (like $2^x = 8, x=3$)
What if I don't? For example, how do I solve for $x$ ...
2
votes
1answer
45 views
Interpolation between iterated logarithms
I am investigating the family of functions $$\log_{(n)}(x):=\log\circ \cdots \circ \log(x)$$
Is there a known smooth interpolation function $H(\alpha, x)$ such that $H(n,x)=\log_{(n)}(x)$ for ...
4
votes
1answer
109 views
Solving a transcendental equation consisting of a quadratic part and a part involving inverse Lambert W functions
Question statement
I would like to solve the following equation in the two variables $x$ and $y$:
\begin{gather}
0 = x^2 - a y^2 + i b [x y - W^{-1}(x)W^{-1}(y)] ,
\end{gather}
where $a$ and $b$ are ...
0
votes
2answers
47 views
Why does this inequality hold: $4n+2\le4n\log{n}+2n\log{n}$
Why is the following true? (I came across this in an algorithm analysis book but this inequality is not related to algorithm analysis)
$$
4n+2\le4n\log{n}+2n\log{n}
$$
4
votes
4answers
84 views
Solve these equations simultaneously
Solve these equations simultaneously:
$$\eqalign{
& {8^y} = {4^{2x + 3}} \cr
& {\log _2}y = {\log _2}x + 4 \cr} $$
I simplified them first:
$\eqalign{
& {2^{3y}} = ...
2
votes
3answers
69 views
A question on log.
What is $\log{(1)}$ to the base of $1$?
My teacher says it is $1$. I beg to differ, I think it can be all real numbers! i.e., $1^x = 1$, where $x\in \mathbb{R}$.
So I was wondering where I have gone ...
-1
votes
1answer
57 views
$\log$ transform of the fundamental theorem of arithmetic? [closed]
Taking the canonical form of the fundamental theorem of arithmetic in the form:
$$n=\prod_{j=1}^\infty p^{m_j}_j \qquad ;m_j\in \Bbb N_0$$
Does anybody know about a $\log n$ transform of this?
Note: ...
3
votes
1answer
46 views
1
vote
4answers
69 views
Solve this equation $\log_2x=\log_{5-x}3$
Solve this equation
$$\displaystyle \log_2x=\log_{5-x}3$$
the answer is $x=2,x=3$
http://www.wolframalpha.com/input/?i=log_2%28x%29%3Dlog%285-x%2C3%29
Can you give me some hint
3
votes
1answer
57 views
integrating logarithm or x raised to a power?
$\int\frac{15}{x}dx$ would be 15$\int\frac{1}{x}dx$ = $15\ln|x|+c$.
This seems like a silly question but I'm feeling exceptionally dense today. Why would you apply the logarithm rule, why wouldn't ...
2
votes
1answer
40 views
determining sign of function containing logarithm.
I would like to know the sign of the following term in general. I tried approximately and it was negative. Is there any $m_0$ such that for all $n>m>m_0$, the following function is positive or ...
0
votes
2answers
40 views
Arithmetic with the natural log
We have:
$$ \ln(p^3 + 4) - \ln(4) = 2$$
What I did is:
$$ \ln (p^3 + 4) = \ln(4) + \ln(e^2)$$
$$p^3 + 4 = 4 + e^2$$
$$ p = e^{2/3}$$
Why is this incorrect?
4
votes
1answer
124 views
How do you solve this equation: $10 = 2^x + x$?
Is it possible to solve this equation?
\begin{align}
a &= b^x + x \\
a-x &= b^x \\
\log_b(a-x) &= x
\end{align}
If $a$ and $b$ are known, how do you find $x$?
-2
votes
0answers
20 views
mortgage total PMI calculation [closed]
Suppose I took a loan amount of 250000 of home value of 300000 with fixed PMI (Primary Mortgage Insurance) of 0.5%. I am able to calculate the PMI per month but how to calculate total PMI paid.
PMI ...
0
votes
3answers
42 views
$\log_x(y)=m^4$ and $\log_y(x)=5/(m^3)$, what is the value of $m$?
$\log_x(y)=m^4$ and $\log_y(x)=5/(m^3)$, what is the value of $m$?
1
vote
1answer
39 views
Making a logarithmic equation that starts at $(0,0)$ and passes through $(x, y)$?
I'm writing a computer program and for fading sound, it's best to do it in a logarithmic equation. What I need it to find a graph of the "volume" that starts at (0, 0) [x is the time, y is the volume] ...
0
votes
0answers
43 views
Expressing $\int _0^1da\int _0^{1-a}\ln ((a-1+b)^2-4 y a b )+\int_0^1 da\int _0^{1-a}\frac{(a-1+b)^2}{(a-1+b)^2-4 y a b}db$ in terms of dilogarithms
I came across these integrals and I'm trying to rewrite them in terms of Dilogarithms: $\mathrm{Li}_2(z):=-\int_0^z \frac{\mathrm{d}s}{s}\log(1-s)$. Can anyone suggest how to contunue? If there is a ...
6
votes
6answers
173 views
Elegant way to solve $n\log_2(n) \le 10^6$
I'm studying Tomas Cormen Algorithms book and solve tasks listed after each chapter.
I'm curious about task 1-1. that is right after Chapter #1.
The question is: what is the best way to solve: ...
