Questions related to real and complex logarithms.

learn more… | top users | synonyms

0
votes
2answers
76 views

Evaluate $\frac{d}{dx}\{(\sin x)^{\cos x} + (\cos x) ^{\sin x}\}$ with logarithmic differentiation

Spivak asks us to evaluate $$\dfrac{d}{dx}\{(\sin x)^{\cos x} + (\cos x) ^{\sin x}\}$$ by logarithmic differentiation. Does he mean for us to evaluate each term separately (which seems to turn out to ...
4
votes
6answers
98 views

$\lim_{n\to\infty}\left(1+\frac{3}{n}\right)^\frac{n}{2}$

I am trying to resolve this to number $e$. However, I would like to do it in the simplest form. just a note I already tried wolfram but I would like someone to give me a simpler solution. ...
-3
votes
2answers
103 views

Prove that $e ^ π$ > $π ^ e$. [duplicate]

Prove that: $$e ^ π > π ^ e.$$ Hint: Take the natural log of both sides and try to define a suitable function that has the essential properties that yields the above inequality
10
votes
2answers
238 views

$\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$

Prove that: $\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$, for $n > 1,\, n\in \mathbb{N}$ For example. For $n=2$, we have $\lfloor ...
0
votes
0answers
28 views

Why can't the base of a logarithm be negative? [duplicate]

I understand why the base of a logarithm can't be 0 or 1, but why negative? What I found out is that when the base is negative we get imaginary results when the powers are rational numbers with odd ...
1
vote
2answers
120 views

Why must the base of a logarithm be a positive real number not equal to 1?

Why must the base of a logarithm be a positive real number not equal to 1? and why must $x$ be positive? Thanks.
0
votes
1answer
24 views

Expected Value of Exponential

I want to calculate $\log E[\exp(-\sqrt{d} S \epsilon)]$, where $\epsilon \sim N(0,1)$ and everything else is deterministic. The result should be $\frac{d}{2}||S||^2$ but why?
0
votes
1answer
45 views

Prove $\displaystyle n\log\left(1+\frac{1}{n}\right) - \log\left(1+\frac{1}{n+1}\right) < \log\left(1+\frac{1}{n+1}\right)$

I'm trying to prove the above inequality, assuming $n\ge1$. I've been working on this one using log properties and trying to reduce this inequalitiy to simpler ones. Though!, is it even correct? or am ...
0
votes
1answer
28 views

Numerical Analysis - show something about the rate of convergence

We are given an iterative method for finding roots, $x_{n+1}=g(x_n)$, we are given the rate of convergence of this method is $p$, and also that: $$\lim _{n \to \infty} \frac{|e_{n+1}|}{|e_{n}|^p} = ...
2
votes
1answer
153 views

Find the volume of the solid obtained by rotating the region bounded by $y = ln x$, $y = 0$, $x = 2$ about the $x$-axis

I have the problem: Assuming $y = ln(x)$, and $y = 0$, find the volume bound by these two lines and the point $x = 2$ if the area were rotated around the $x$-axis. I ended up with $2\pi\int_1^2 ...
3
votes
1answer
53 views

Difficult Integral Involving the $\ln$ function

Please help me solve this integral! I have tried multiple different procedures for integration by parts, as well as substitution and have not come up with anything. $$\int\frac{\ln x}{(\ln x+1)^2}dx$$ ...
2
votes
1answer
20 views

What is the number of real roots of $(\log x)^2- \lfloor\log x\rfloor-2=0$ $\lfloor\,\cdot\,\rfloor$ represents the greatest integer

Question : What is the number of real roots of $(\log x)^2- \lfloor\log x\rfloor-2=0$. $\lfloor\,\cdot\,\rfloor$ represents the greatest integer function less than or equal to $x$. I know how to ...
2
votes
2answers
27 views

Let $x_0 > x_1 > x_2>x_3$ be any positive real numbers . What is the largest value of the real number k such that..

Question : Let $x_0 > x_1 > x_2>x_3$ be any positive real numbers . What is the largest value of the real number k such that $$\log \frac{x_0}{x_1}1993 + \log \frac{x_1}{x_2}1993 +\log ...
1
vote
1answer
23 views

Comparing logarithmic functions. Master Method

I'm learning the master method and am looking for help on how to best approach comparing two functions asymptotically. More specifically, I have: ...
0
votes
1answer
37 views

show that $(1+ \frac {x}{n})^n < e^x$ and $e^x < (1- \frac{x}{n})^{-n}$ if $x<n$

If $n$ is a positive integer and if $x>0$,show that $(1+ \frac {x}{n})^n < e^x \quad$ and that $\quad e^x < (1- \frac{x}{n})^{-n} \quad $ if $x<n$ I proved the first one by the ...
1
vote
2answers
41 views

Solving equations with logarithms

I'm having trouble with solving equations that has logarithms in them. For example: $$x^{\log(x)} = \frac{100}{x}$$ How can I solve this? I have reed about how to do it but when I try to do the same ...
1
vote
1answer
29 views

Differentiating logarithms

I am trying to prove that $$ f(x) = ^alog(x) => f'(x) = \frac {1} {ln(a)*x} $$ So I start at$$ f(x) = ^alog(x) $$ Then I move to:$$ f(x) =\frac {ln(x)} {ln(a)} $$ And there I get stuck: I want ...
1
vote
1answer
63 views

How do computers calculate the log of a value? [duplicate]

I'm not sure if this question belongs on StackOverflow or here (please let me know if the former, and i'll delete this and ask there), but I was wondering how the ...
-1
votes
1answer
20 views

Rewrite a formula in terms of exponential to the power of logarithm

I would like to rewrite the following formula, f(x). how can I rewrite the f(x) $$ f(x) = ...
2
votes
1answer
162 views

Logarithms melting my brain

So I've got an inequality: $\ln(2x-5) > \ln(7-2x)$ and I attempt to solve by doing the following: $$\frac{\ln(2x)}{\ln(5)} > \frac{\ln(7)}{\ln(2x)}$$ $$\Rightarrow \ln(2x) \cdot \ln(2x) > ...
14
votes
2answers
978 views

Show that these two numbers have the same number of digits

I want to show that for $n>0$, $2^n$ and $2^n + 1$ have the same number of digits. What I did was I found that the formula for the number of digits of a number $x$ is $\left ...
1
vote
2answers
39 views

Need help with a proof concerning zero-free holomorphic functions.

Suppose $f(z)$ is holomorphic and zero-free in a simply connected domain, and that $\exists g(z)$ for which $f(z) =$ exp$(g(z))$. The question I am answering is the following: Let $t\neq 0$ be a ...
2
votes
2answers
248 views

Why can't I use product rule to derive x ln(3)?

The product rule is defined as $$(f \cdot g)' = f' \cdot g + g' \cdot f.$$ I have the following function $u(x) = x\cdot \ln(3)$. I understand that you can derive it by implicit differentiation and ...
0
votes
2answers
36 views

general formula for $\log_x(y)$ when $y$ is negative

I'm looking for a general formula for solving a problem of the form $\log_x(y)$ when $y<0$. It seems like the formula is $\frac{\ln(|y|)+\pi i}{\ln(x)}$, but I would like to know how this is ...
1
vote
2answers
45 views

how to find the integral of a rational logarithmic function

I can't seem to figure this one out, it is: $$\int\frac{\ln(x)}xdx $$ I substituted $u$ for $\ln(x)$, so $u = \ln(x)$ and $du = \frac1x dx$ then to find $x$ in terms of $u$: $e^u = x$ so ...
1
vote
3answers
105 views

Solve the equation $\log(z^2-1)=i\pi/2$

I set $z=x+yi$, so: $$ \log[(x+yi)^2-1]=\log(x^2+2xyi-y^2-1)=\log (r+iθ)=i\pi/2$$ than I get $x^2-y^2=1$ and I have no idea how to continue.
1
vote
3answers
65 views

How is the Logarithm derived from the exponential function? (aren't they inverses?)

I've been learning logs in school, and my teacher, friend, and I are stumped on something. How does one derive the logarithmic function from the exponential function? My friend thinks Tayler Series ...
1
vote
1answer
42 views

Logarithm rules for complex numbers

Are the logarithm rules true for complex numbers? We know that for positive real numbers $a$, $b$, $c$ and real number $d$ that: $$\log_b\left(a^d\right)=d\log_b(a)$$ $$\log_b(a) = ...
0
votes
3answers
56 views

$\ln( \exp( \ln( \exp( 64 )^{1/2} )^{1/2} )^{1/2} )$

I keep getting the answer 8. But the textbook as well as wolfram say it's 8^(1/2) or in other words 2(2^(1/2)). Here are the steps I took, basically just following the rules of logarithms. ...
0
votes
5answers
41 views

Express $\log_{3}5$ in terms of $p$ and $q$.

If $p=\lg5$ and $q=\log_{3}2$, express $\log_{3}5$ in terms of $p$ and $q$. Um really confused! How do I solve this?
1
vote
2answers
34 views

If $p=\lg 5$, express the following in terms of $p$.

If $p=\lg 5$, express the following in terms of $p$. Express $\log_{5}2$ in terms of $p$. =$\log_{5}2$ =$\frac{\log_{10}2}{\log_{10}5}$ =$\frac{\log_{10}2}{p}$ Then how to simplify it? My book's ...
0
votes
2answers
46 views

Having trouble solving an inequality

I'm a trying to prove a recurrence relation (by substitution) for an algorithm class and I'm shamefully stuck in a rather simple looking inequality. I need to solve this inequality for constant $c$: ...
1
vote
1answer
19 views

Comparing sum of fixed rate value to sum of escalating value

Find the number of years, $n$, until the sum of an escalating value/income exceeds the sum of a higher fixed level value/income. Income fixed at £8405.64 Income escalating @ 3% per annum from ...
1
vote
2answers
61 views

What is the domain of this function? (Don't know how to solve it, logarithms…)

Please explain how you solved it, thanks. $f(x)=\sqrt{\log_x2 - \log_2x}$
1
vote
2answers
55 views

Changing base of a logarithm by taking a square root from base?

From my homework I found $$\log_9{x} = \log_3{\sqrt{x}}$$ and besides that an explanation that to this was done by taking a square root of the base. I fail to grasp this completely. Should I need to ...
0
votes
1answer
24 views

Evaluate $\log_{5}19^2 / (\log_{3}15)$

Evaluate $(\log_{5}19^2)/(\log_{3}15)$ I did this, but I get $1.484$, whereas it's $1.038$ in my book. I used the method: $\log_{a}b = \frac{\log_{c}b}{\log_{c}a}$
0
votes
0answers
14 views

compounding interest question

A bank advertises that it compounds interest continuously and that it will double your money in 7 years. What is the annual interest rate? P(t) = P*e^kt P(t)/P =2 e^7k = 2 take ln of Both Sides ...
1
vote
2answers
24 views

Continuous compounding question

A population of rabbits starts out with $100$ rabbits. The growth rate is $11.7$% per day. Determine the exponential equation. Is it $$\mathbb {P(t)} = 100e^{11.7t}$$ Can you guys give me the ...
0
votes
0answers
30 views

Why to Use Logatihm Function Instead of Multiplication at Comparison?

It is obvious that: if x < y than log(x) < log(y) so if we see an equation as like that: a * b we write it as ...
17
votes
2answers
473 views

Is this function a constant?

I am a french guest and I hope that my english isn't too bad... So here is my issue : I consider an entire function $f$ which satisfies the following property for all complex number $z\in \mathbb{C}$ ...
1
vote
2answers
34 views

Taking the logarithmic derivative of an exponential difference function after applying L'Hospital's Rule

Can somebody please explain the following application of L’Hospital’s Rule? Find the limit: $$\lim_{x\rightarrow 0} \frac{5^x-3^x}{x}$$ Solution: Determining that this function has indeterminate ...
1
vote
3answers
38 views

Value of $2^{-1-i}$ in the complex

I am trying to find the value of $2^{-1-i}$. I rewrite it like this, $2^{-1-i}=e^{\ln(2)(-1-i)}={1\over{e^{\ln2}e^{i\ln2}}}=1/2$ Since $e^{i\ln2}=e^{Re(i\ln2)}=e^0=1$. This looks way nicer than ...
1
vote
0answers
21 views

Mapping a deleting ray to a horizontal strip

So, this is my question: D is a domain obtained by deleting the ray $x\leq 0$. And $G(z)$ is a branch of $log(z)$ on $D$. I want to show that G maps D onto a horizontal strip of width $2\pi$. Show ...
0
votes
0answers
26 views

Suppose y is a prime and that b has order 3 modulo y. What is the order of b+1? [duplicate]

order is 6 by showing the expansion of (b+1)^6, but are there other solutions where I have to solve for all the pairs (y,b) that meets the conditions of the problem ?
0
votes
1answer
37 views

Exponential of $\bar{z} $

I am currently reading the book Complex Variables by Stephen Fisher, there is one paragraph that was written like this: Establishing the following relation, and they write ...
0
votes
0answers
36 views

Calculate $\lceil \frac{n}{log_2k} \rceil; n \geqslant 1, k \geqslant 2$ with only integer functions

How to calculate following expression with only integer fuctions? $$\lceil \frac{n}{log_2k} \rceil; n \geqslant 1, k \geqslant 2$$ I mean with using of only integer division, integer log with base ...
1
vote
2answers
77 views

What are the basic, fundamental concepts of logarithms?

Soon I will be learning of logarithms, but I wish to have a head start over my class mates or at least a fair greeting to the concept. I would appreciate a very clear-cut answer with very straight to ...
5
votes
2answers
518 views

U-substitution for integral of 1/(1+e^x)dx. What am I doing wrong?

Here is my work, witth the right answer. I feel like every step is right, but somehow I am getting the wrong answer. How? $$ \int \frac{1}{1+e^z}dz = \int\frac{1}{e^z(\frac{1}{e^z} + 1)}dz ...
0
votes
6answers
40 views

Logarithmus to simple subtraction - how?

I am learning for a math exam and have the following solution: $$ 0.01 = 0.5^n\\ n \cdot \log 0.5 = \log 0.01\\ n=\frac{\log 0.01}{\log 0.5} $$ OK, so far, so good. (I guess) But now, it gets ...
2
votes
1answer
79 views

Solve exponential-polynomial equation

Solve the equation in $\mathbb{R}$ $$10^{-3}x^{\log_{10}x} + x(\log_{10}^2x - 2\log_{10} x) = x^2 + 3x$$ To be fair I wasn't able to make any progress. I tried using substitution for the ...