Questions related to real and complex logarithms.

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-3
votes
0answers
37 views

Help solving a Problem [on hold]

The population of a particular city (in thousands) can be modeled by the function $$P(t)= \frac{500}{1+20e^{-0.05x}}$$ where x is the number of years after 1920. In what year was the growth rate of ...
0
votes
1answer
14 views

Is $argmin_{\mathbf{x}} f(\mathbf{x})=argmin_{\mathbf{x}} \log{f(\mathbf{x})}$ always true?

Assuming $\mathbf{x}\in \mathbb{R}^n$, $f(\mathbf{x})\gt0 \forall\mathbf{x}\in\mathbb{R}^n$, is $argmin_{\mathbf{x}} f(\mathbf{x})=argmin_{\mathbf{x}} \log{f(\mathbf{x})}$ always true? Why?
0
votes
0answers
33 views

Can't find solution to Calculus 8th (Adams, Essex) problem

I've been sitting here for hours trying to find a solution to his problem. If you have the function $g(y)$, which is the inverse of $f(x) = x^x,\\ e^{-1} \leq x < \infty,$ show that ...
2
votes
5answers
103 views

How I could show that :$\log1=0$?

I would be like somone to show me or give me a prove for this : Why $\ln 1=0$ ? Note that $\ln$ is logarithme népérien, the natural logarithm of a number is its logarithm to the base $e$. Thanks ...
0
votes
0answers
11 views

Natural logarithm of a square matrix without eigen-analysis

I'm trying to find a method to determine the natural logarithm of a square nonsingular matrix without using eigenvalues or eigenvectors. So far, I've only found this method: ...
2
votes
0answers
67 views

${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$

I have found the following new result connecting to rational log-cosine integrals. Proposition. \begin{align} \displaystyle & {\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} ...
5
votes
1answer
79 views

Irrational to power of itself is natural

I've been thinking about a natural number like $n$ so that $x^x=n$ for some irrational $x$ but i couldn't find anything. As i didn't know how to approach the problem at all, i tried to make some ...
0
votes
2answers
27 views

Value of the logarithmic expression [on hold]

Why $\log_6 \sqrt{6} = 1/2$? I know that $\sqrt{6} = 6^{1/2}$.
1
vote
4answers
31 views

Find the value of the Logarithmic Expression

Why is $\log_6 1$ equal to $0$ ?
1
vote
3answers
29 views

Solving for the value inside a base 10 logarithm

I have an equation of $\log(d)=(-x-A)/(10n)$ that I need to solve for $d$. How do I "reverse" the logarithm to obtain $d$? I apologize if this is super easy, I just can't even figure out how to Google ...
0
votes
0answers
23 views

Prove natural log between two finite harmonic sums [duplicate]

Prove for n in the naturals we have: $$\sum_{k=2}^n 1/k \le \ln(n) \le \sum_{k=1}^{n-1} 1/k$$ Intuitively this makes sense to me but I can't for the life of me figure out how to start this proof.
0
votes
0answers
28 views

Calculating log and trigonometric functions using only +,-,/,*

How to calculate logarithm and trigonometric functions (sin, cos etc.) on base n with using +,-,/,* ? Is there any way to do it?
2
votes
5answers
434 views

I have issue in calculating log values

How is $$log_42= \frac{1}{2}$$ ? Any formula to how we calculate this? I know i am confused when base is larger digit than log value term.
1
vote
2answers
65 views

Can someone suggest a function to achieve this graph?

I need a function that achieves a curved, logarithmic (I think) graph. The graph plots the level progression of a player of a simple game over time. Some players might progress faster, others slower, ...
0
votes
0answers
40 views

solving the logaritham [duplicate]

I was trying to solve: $\log_2 x \cdot \log_{0.25} x \cdot \log_{0.125} x \cdot \log_{16} x > \frac {2}3$ heres my attempt at it; using logaritham laws and a little algebra we get from $\log_2 x ...
0
votes
1answer
23 views

solving equation (indices/logarithms)

I don't really understand the Logarithms concept when it comes to question with ln or log with base e.for example question like this: 1.solve the equation $$x^4\mathrm{e}^{-2\ln(x)}=18-3x$$
2
votes
1answer
94 views

Calculate $I= \int_{1}^{e}\frac{(1+\ln x)x}{(1+x\ln x)^2}dx$

Please help me solve this: (level = high school) $$ \int_{1}^{e}\frac{(1+\ln x)x}{(1+x\ln x)^2}\,dx $$ Thanks
0
votes
1answer
23 views

What is the minimum degree of x so that it is greater than or equal to ln(x)?

I was thinking of this question and couldn't find it anywhere. I was trying to find a solution by finding the maximum of the function n = ln(ln(x))/ln(x) but I'm not sure if that's gonna work. Thanks ...
1
vote
1answer
35 views

Why does the same inequality give different answers?

$\left(\log _2\left(x\right)-2\right)\left(\log _2\left(x\right)+1\right)<0$ has a solution $\frac{1}{2}<x<4$ But when we take the second part alone that is $\left(\log ...
0
votes
1answer
37 views

Choosing a branch of the square root

Assume $O$ is the compliment of the non-positive part of the real line to the complex plane. This is an open and connected set. Only one of the values of $\sqrt z$ in $O$ has positive real part. With ...
1
vote
3answers
54 views

Convergence N'th Harmonic number minus the Natural Logarithm of N. [duplicate]

I was hoping if someone could show me the proof of exactly why this converges to the Euler–Mascheroni constant.
0
votes
3answers
87 views

How can I solve the following equation? [closed]

$\log_2{\frac{x-3}{x+2}}≤0$ Thank you.
-2
votes
2answers
111 views

How to show $\log_2 x \cdot \log_{0.25} x \cdot \log_{0.125} x \cdot \log_{16} x > \frac {2}3$?

I was trying to solve $\log_2 x \cdot \log_{0.25} x \cdot \log_{0.125} x \cdot \log_{16} x > \frac {2}3$ and I keep getting a partial answer of $x>4$ though answer key suggests a more expanded ...
0
votes
2answers
33 views

Simultaneous log equations

I'm going through logarithms at the moment, and I can't solve this simultaneous equation: $$\log x - \log 2 = 2\log y$$ $$x - 5y + 2 = 0$$ I've tried substituting both $x$ and $y$ to no avail: ...
2
votes
2answers
45 views

solving equations with powers

Im trying to solve the equation $$3\cdot2^{-2/x} + 2\cdot9 ^{-1/x} = 5\cdot6^{-1/x }$$ So far I tried applying logaritmas but it didnt prove helpful...are there any other ways?
1
vote
1answer
22 views

Using Stirling's approximiation to show that $(\log(\log n))!$ is $O(n^k)$

I am trying to show the following: Prove, using Stirling's approximiation, that $(\log(\log n))!$ is $O(n^k)$ for some positive constant $k$. Stirling's approximation is $$n!=\sqrt{2\pi ...
3
votes
1answer
67 views

Why $\ln(1)\neq 2\pi ik$

Given that $e^{2\pi ik}=1$ for all $k \in \mathbb{Z}$, why isn't $\ln{e^{2\pi ik}}=2\pi ik$? On the other hand $\ln(1)=0$. What am I missing here?
0
votes
1answer
46 views

How to simplify this equation? $1+\sqrt {2^{2a_{n}+b_{n}+1}-16^{a_{n}}-4^{b_{n}}}=\log _{3}\left( a_{n}+b_{n}\right) $

How to simplify this equation? $1+\sqrt {2^{2a_{n}+b_{n}+1}-16^{a_{n}}-4^{b_{n}}}=\log _{3}\left( a_{n}+b_{n}\right) $
1
vote
0answers
32 views

Find $\dfrac {a_{1}}{b_{1}+1}-\left( \dfrac {a_{1}}{b_{1}+1}\right) ^{2}+\left( \dfrac {a_{1}}{b_{1}+1}\right) ^{3}-…$ [closed]

Define $a_{n}$, $b_{n}$ be the sequence of a positive real number and be consistent with $1+\sqrt {2^{2a_{n}+b_{n}+1}-16^{a_{n}}-4^{b_{n}}}=\log _{3}\left( a_{n}+b_{n}\right) $. Find $\dfrac ...
0
votes
1answer
17 views

Solving Logarithms involving ceiling function

I need to solve the equation $\lceil \log_B(M) \rceil = S$ for $B$ when $M$ and $S$ are known, $M$ and $S$ are integers, and $B < M$. Were the ceiling function not there, it would be trivial, ...
2
votes
1answer
36 views

Name for a Logarithm Identity/Property

I came across a neat logarithm fact today: $\large n^{\log_bx} = x^{\log_bn}$ One simple proof is: $\large \log_bx\cdot \log_bn=\log_bx\cdot \log_bn$ $\large \Rightarrow ...
5
votes
5answers
63 views

What is the limit of $\log_k(k^a + k^b)$ for $k \to +\infty$?

I'm not very good with analysis (I never studied it) but because of my "work" on other topics of mathematics I came to this problem. $$\lim_{k \to +\infty }\log_k(k^a + k^b)=\max(a,b)$$ I'm sure ...
0
votes
3answers
61 views

Solve logarithmic equation: $2\log_7 (x+2) - \log_7 (3x+10) = 0$ [closed]

Please, can someone check if this is the right answer $$x= -2 \pm \sqrt{3x + 10}$$ Thank you.
2
votes
3answers
105 views

A series converging (or not) to $\ln 2$

I have come across the following series, which I suspect converges to $\ln 2$: $$\sum_{k=1}^\infty \frac{1}{4^k(2k)}\binom{2k}{k}.$$ I could not derive this series from some of the standard ...
3
votes
4answers
53 views

A general definition of Entropy (i.e. may or may not be expectation of the Log of the probabilities) [closed]

Entropy may be defined as Entropy = Σ G(p(x)) Where 'G' is any function that goes asymptotically to plus infinity as it approaches zero from the positive side and is monotonic between 0 and 1 ...
0
votes
2answers
49 views

Proof the expession $\log_{12}{18}*log_{24}{54} + 5(\log_{12}{18}-log_{24}{54})=1$

I am trying to proof the following expression (without a calculator of course). $\log_{12}{18}*\log_{24}{54} + 5(\log_{12}{18}-\log_{24}{54})=1$ I know this isn't a difficult task but it's just ...
1
vote
2answers
49 views

Definition of $a^b$ for complex numbers

Problem statement Let $\Omega \subset C^*$ open and let $f:\Omega \to \mathbb C$ be a branch of logarithm, $b \in \mathbb C$, $a \in \Omega$. We define $a^b=e^{bf(a)}.$ $(i)$ Verify that if $b \in ...
2
votes
1answer
36 views

How to solve this logarithmic equation?

I want to solve this equation: $$8n^2 = 64n\log_{\ 2}(n)$$ After some steps, I get to a point in which I believe, the only way to proceed is to apply something like Bolzano's or Newton's method to ...
2
votes
2answers
49 views

I need help on the process of solving this derivative.

How do I go about solving this derivative. $$f(x)=\ln\left(\frac{7x}{x+4}\right)$$ I go from this to $$1. \quad f(x)=\ln(7)+\ln(x)-\ln(x+4)$$ and then $$2. \quad f'(x)=\frac{1}{x}-\frac{1}{x+4}$$ then ...
2
votes
2answers
81 views

How do I simplify $\log (1/\sqrt{1000})$?

How do I simplify $\log \left(\displaystyle\frac{1}{\sqrt{1000}}\right)$? What I have done so far: 1) Used the difference property of logarithms $$\log ...
0
votes
2answers
41 views

Help me solve this…

Assuming $a=\log 2$ and $b=\log 3$ (log is the base 10 logarithm). I have to find $\log_5 288$. How can I do this? Edit: I've tried transforming $\log2$ to $\frac{\log_5 2}{\log_5 10}$ and same for ...
5
votes
3answers
163 views

Help with logarithmic definite integral: $\int_0^1\frac{1}{x}\ln{(x)}\ln^3{(1-x)}$

I'm look for a closed form evaluation of the following improper definite integral involving logarithms: $$\begin{align} I:&=\int_{0}^{1}\frac{1}{x}\ln{(x)}\ln^3{(1-x)}\,\mathrm{d}x\\ ...
0
votes
0answers
23 views

Using math functions to time finales of a fireworks show

This year, I have the honor of programming two finales for a fireworks show. I want to use math. I suspect that I should use a function such as square root or log to specify the decreasing pause ...
0
votes
1answer
11 views

Consumption change calculation

I want to calculate yearly consumption change according to the following formula: $$C_{t+1}=C_{t}e^{x_{t}}$$ I need to calculate ${x_{t}}$. I have the consumption data $C_{t+1}$ and $C_{t}$.
-3
votes
2answers
47 views

How to get this answer [closed]

Anyone help me solve this question $$\ln u + 2 \ln(1-u) - 2 \ln(1+u) = 2 \ln x + \ln c$$ I have the answer as $\frac{x y}{ (x^2 - y^2)^2} =c$, but I cant figure out how get this answer.
2
votes
1answer
29 views

Does $\sum_{i=1}^{k-1}\lceil \log_2\frac{N}{i}\rceil$ have a closed form?

Does the following have a closed formula? $$\sum_{i=1}^{k-1}\left\lceil \log_2\frac{N}{i}\right\rceil$$
7
votes
3answers
182 views

Is $ln(x)$ ever greater than $x$

Is $\forall x \in \mathbb{R}, \ln(x) \lt x$ a true statement? Just wondering for some convergence related thing
2
votes
1answer
70 views

Checking derivation of y = a^x

Can you tell me if there are any flaws with this derivation of $y = a^x$... The assumptions are that the derivative $$\frac{d}{dx}e^x = e^x$$ and that the derivative $$\frac{d}{dx}\ln x = ...
0
votes
1answer
42 views

Is there a property for log(n)/n?

I found a small exercise which I couldn't figure what to do, so I found a solution. Then I tried to understand it and everything went well until I got to this part: $$\frac{1}{8} = ...
0
votes
2answers
49 views

Help me to solve math homework on logarithmic

How to solve this math home work? Please help.. What is the value of $\log \left(\dfrac{i\pi}{2}\right)$ ? I got to know the answer is "$\dfrac{i\pi}{2}$", but don't know how to solve it. Please ...