Questions related to real and complex logarithms.

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0
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2answers
30 views

Rewrite a Logarithm as a product of logarithms

Can anyone help me to understand this? $$\log_2 n = \log_2e \log n$$
-2
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1answer
43 views

Difficulty finding point of inflection

The Problem is... $N(t)=\frac{200,000}{1+999e^{-0.4t}}$ Use logarithmic differentiation to find the time to the point of inflection. I know that in order to find the point of inflection I must set ...
1
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3answers
15k views

What is log ? What does it mean? How does it transform a number?

What is log ? What does it mean? How does it transform a number? /I'm a code and see log being used in legacy code I have to change. Thanks
1
vote
1answer
33 views

Limit with logarithm: $\lim_{n \to \infty} \frac{n^\alpha}{\ln^\beta n}$

What is the limit $\lim_{n \to \infty} \frac{n^\alpha}{\ln^\beta n }$ (ln=natural logarithm) for alfa real and less than zero? I found out it is zero for $\beta\ge0$, since then you can use the ...
4
votes
3answers
991 views

Lambert- W -Function calculation?

I have an equation of the form: $$ n \log n = x $$ Upon searching I came across the term "Lambert- W -Function" but couldn't find a proper method for evaluation, and neither any computer code for ...
0
votes
1answer
40 views

Please explain the logarithmic equation

The default equation is $(1 + x)^3=4^{-y^2} $ I solved as follows: $(-3\log_4(1+x))^{1/2}=y$ With the logarithm base equal to $4$, my idea is that $4$ is the number we have in the right part of ...
0
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3answers
43 views

Sieve of Eratosthenes Time Complexity Clarification

I've found plenty of sources claiming that the time complexity of the prime sieving algorithm Sieve of Eratosthenes is $O(n\log(\log n))$ where $n$ is the input. However, is this $\log_{10}$ or $\ln$? ...
1
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2answers
71 views

$2\ln(-x) \neq \ln(-x^2) ? $

I know the rule $$n\ln(x) = \ln(x^n)$$ But this doesn't apply to $$2\ln(-x) = \ln(-x^2)$$ Can you see what I'm not understanding?
1
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4answers
49 views

How to prove that $(x+c)\log(\frac{c+x}{x})>c$

How to prove that $(x+c)\log(\frac{c+x}{x})>c$ for $x, c > 0$? For $\frac{c+x}{x} \ge e$ it's obvious.
0
votes
1answer
19 views

When is a function a branch of another multi-valued function?

Definition (Branch): A branch of a multiple-valued function $f$ is $\color{teal}{\text{any}}$ single-valued function $F$ that is $\underline {analytic}$ in some domain at each point $z$ ...
1
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0answers
12 views

Financial Mathematics--Finding Compounding Period given Annual and Effective Interest Rates

I'm trying to find a compounding period C when given an annual interest rate r and effective annual yield i. I'm working with the following equation: $i=(1+r/C)^C-1$ I'm having trouble re-writing ...
1
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3answers
35 views

Limit of $f(x)=|\log x|$

My textbook solved this problem: Find $f'(1^{-})$ if $$f(x)=|\log x|$$ for the interval $x>0$ The textbook solved it by using the method described below: $$f'(1^{-})=\lim\limits_{x\to 1^{-}} ...
0
votes
0answers
26 views

Simplify/expand $\ln \left(\sum^n_{i=1}x_i^{\theta-1}\right)$

Can someone help simplify/expand this natural log? I want to bring the $\theta - 1$ down in front of the $\ln$ but I don't know how the rules work with the summation. $$\ln\left( ...
1
vote
0answers
75 views

Hard Logarithm Integral [duplicate]

I recently had this integral in my homework and don't really know how to proceed. $\int^1_0\frac{\ln (1+x^{2+ \sqrt3})}{x+1}dx$ So far I figured that $ \int^1_0\frac{\ln (1+x^{a})}{x+1}dx= ...
4
votes
1answer
95 views

How to compute $\int 1/x \, dx$ without knowing its anti-derivative

How do you compute $$\int\frac 1x \, dx$$ without knowing its anti-derivative to start with? Is there a way to do it by parts or substitution?
1
vote
0answers
36 views

Really simple question: Add $\bar4.74628$ and $ 3.42367$ .I just need to cross check answer.

Add $\bar4.74628$ and $ 3.42367$ This question is about characteristics and Mantissa.I thought my book has written the wrong answer in the example.I just wish to cross check because this seem like ...
2
votes
2answers
28 views

Natural logarithm power notation

I am trying to understand how to use Dirichlet's test for convergence and saw an example here (example 2). Show that $\displaystyle\sum_{i=1}^\infty \frac{2^{2n}n^2}{e^n\,n!}\frac{1}{\ln^2n}$ ...
0
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1answer
32 views

Adding logarithms with different bases

Just had an exam, this sinister question I know I did wrong lingers in my mind: Solve for $x$, $$2-\log_3(x-7) = \log_{\frac{1}{3}} (2x)$$ On phone not sure how to write the equation properly. ...
-1
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1answer
22 views

Big O notaion O(n) and logaritms [closed]

Can someone explain me the subjects Big O notation and logarithms please? I can't understand those subjects For example if I have a question like this: recall that logan is the power to which you ...
0
votes
2answers
54 views

Approximate $\log(1-e^x)$ where $x<0$

The title is pretty self-explanatory, I need to calculate the logit function ($x=\log(p)$): $$x-\log(1-e^x)$$ Where $x<0$, And my problem is to approximate $$\log(1-e^x)$$ I was thinking of ...
1
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1answer
73 views

Logarithm as limiting case of $n$th root

Let $f_n(x) = x^{1/n}$ where $n \in \mathbb N$, and let $g(x) = \log(x)$. We can compute $f_n'(x) = \frac{1}{n}x^{-1 + \frac{1}{n}}$ and $g'(x) = x^{-1}$. Let's define $f_\infty(x) = \lim_{n ...
1
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2answers
36 views

Proof of log 2 base 10 value

Is there a way to prove log 2 base 10 <= 0.301 other than verifying the value using a calculator? Please give a detailed explanation, if proof is possible.
0
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2answers
33 views

How to simplify the expression $(\log_9 2 + \log_9 4)\log_2 (3)$

Our test asked to simplify $(\log_9 2 + \log_9 4)\log_2 (3)$. I simplified the first parenthesis to be $\log_9 (8)$. So, now I have $\log_9 (8) \cdot \log_2 (3)$ and I can change to base $10$ and ...
0
votes
1answer
36 views

How to solve the logarithmic equation $\ln(x + 4) = 6$?

I would like to learn the steps for solving this math problem. One of my classmates gave me this problem, and I need help solving it. $\ln(x+4)=6$
-5
votes
1answer
59 views

Proof that $0^0 \neq 1$ [closed]

Suppose that $t = \sqrt{t}^{\sqrt{t}}$, then, it follows that; $$ t^{\sqrt{t}} = \sqrt{t}^{t} \\ \frac{1}{2}t\ln{\left(t\right)} = \sqrt{t}\ln{\left(t\right)} \\ \ln{\left(t\right)}\left[\frac{1}{2}t ...
4
votes
6answers
3k views

Is there an approximation to the natural log function at large values?

At small values close to $x=1$, you can use taylor expansion for $\ln x$: $$ \ln x = (x-1) - \frac{1}{2}(x-1)^2 + ....$$ Is there any valid expansion or approximation for large values (or at ...
1
vote
2answers
85 views

How to solve the equation $x \log \log x = n$

I would like to solve the equation $x \log\log x = n$. I've seen a lot of post about the equation $x \log x$ but here I have a composition of $\log$. How can I solve it ? Thank you very much.
0
votes
1answer
26 views

basic math problem relating to log thought my answer is not matching with the alternative

$1/2 \log c = 0.915$. Calculate $c$. It is a basic math problem but my answers are not matching with the alternatives. $1/2 \log c= c^{1/2}$ $ c^{1/2}= 0.915$ $c = 0.915 \times 0.915=0.83448$ ...
1
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1answer
13 views

Verify whether or not expression is true or not for z>0

I need to verify whether or not the below expression is true or not for $z>0$. I'm trying to understand the rules of logarithms but I can't figure out how to apply it myself or where to even begin. ...
1
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2answers
53 views

Solution of $5^{\log x}+5x^{\log 5}=3$

Solve for $x$ $$5^{\log x}+5x^{\log 5}=3$$ where base of log is $a$, $a>0$ and $a \neq1$ Could someone hint as how to initiate this question? I am not having any idea as how to proceed.
2
votes
1answer
71 views

How to prove $({\log_2 x})^{n+1} \le x^n$

I want to show that $({\log_2 x})^{n+1} \le x^n$ when $n \ge 1$ and $x \ge 1$. I know that ${\log_2 x}$ can be shown to be $\lt x$ with: $x \lt 2^x$ $\log_2 x \lt x$ and obviously adding the same ...
1
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1answer
28 views

How are floating-point numbers logarithmically distributed?

From what I remember from a lecture I had of a course I'm attending called "introduction to computational science", floating-point numbers are distributed logarithmically. What does it mean? And how ...
1
vote
2answers
118 views

Does the series $\sum\limits_{n=1}^{\infty}\dfrac{\ln(\sqrt{n})}{n}$ converge or diverge?

How to determine does this series converge or diverge? I have tried d'Alembert's ratio test but in the limit I get $1$. I suppose I should compare it with some other series, but I can't figure out ...
6
votes
5answers
472 views

Evaluate $\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$ using complex analysis

How do I compute $$\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$$ What I am doing is take $$f(z)=\frac{(\log z)^2}{1+z^2}$$ and calculating $\text{Res}(f,z=i) = \dfrac{d}{dz} \dfrac{(\log ...
7
votes
10answers
421 views

How to prove that $\log(x)<x$ when $x>1$?

It's very basic but I'm having trouble to find a way to prove this inequality $\log(x)<x$ when $x>1$ ($\log(x)$ is the natural logarithm) I can think about the two graphs but I can't find ...
11
votes
2answers
182 views

What is $\int_0^1 \ln (1-x) \ln \left(\ln \left(\frac{1}{x}\right)\right) \, dx$?

There are well-known closed-form evaluations for integrals of the form $\int_0^1 a(x) \ln \left(\ln \left(\frac{1}{x}\right)\right) \, dx $ for certain algebraic functions $a(x)$. For example, an ...
0
votes
1answer
634 views

What is the limit of difference between harmonic series and natural logarithm of n+1?

I'm an undergraduate student in geology and I'm dealing with a project in math. The last question of the project gives me the harmonic series ($A_n = 1 + \frac{1}{2} + ... + \frac{1}{n}$) and this ...
1
vote
0answers
28 views

Nested logarithms to represent real intervals.

Out of curiosity from this question regarding writing real numbers as an iterated sum/difference of square roots I started experimenting with another family of functions: $$\log[ \exp[1] \pm ...
0
votes
1answer
33 views

When will the population of a sample double (using dif-eq)?

I have the initial equation $$\frac{dP}{dt}=kp$$ where P is the population, t is time, and k is some positive constant. The rest of what I'm given is that P(0) = A, what is the time for the population ...
0
votes
2answers
15 views

Solution to initial condition problem

$y=-ln(1-e^{(t+c)})$ I'm trying to find the solution to the initial condition $y(0)=-ln2$ Isolate c $0=ln(2)-ln(1-e^c)$ $0=ln({2\over1-e^c})$ $-e^c=2-1$ $e^c=-1$ $c=0$ I can't figure out ...
0
votes
2answers
53 views

Tricky Logarithmic inequality

I have tried proving this logarithmic inequality but I did not succeed. I tried to put every term on one side, I expanded and tried to use one of the properties of logarithms but the proof does not ...
0
votes
0answers
18 views

Weighted logarithmic ranking

I want to have a ranking of players by percentage of shots made, weighted by the total number of shots attempted. The weighting should follow a log scale, so for example Player A has 100% accuracy, ...
0
votes
1answer
27 views

What is the argument of the logarithm operator called?

In the expression $ln(y)$, what is '$y$' called. I'm asking for a noun analogous to exponent in $x^n$, where '$n$' is called the exponent. If I'm not mistaken, '$x$' in this case is the radix, or ...
1
vote
1answer
16 views

Is it possible to clear the x using the Lambert function?

$ y = \frac{x^2}{4} - \frac{ln(x)}{2} $ Solving, I get to: $ e^{4y} = \frac{e^{x^2}}{x^2} $ But I don't know how to continue.
0
votes
1answer
46 views

Stuck solving $\ln(e^y-1)-y=t+c$ for $y$

I'm trying to solve for $y$ $\ln(e^y-1)-y=t+c$ $e^y-1=e^{(t+c+y)}$ $e^y=e^{(t+c+y)}+1$ $y=t+c+y+1$ Where am I going wrong?
6
votes
0answers
168 views

Are known these identities, that I've deduce using Mobius inversion formula?

I would to know if this formula is right and know (these formula are the same by exponentiation), since I deduce this easily by a standar way (perhaps there are mistakes) using Mobius inversion from ...
4
votes
2answers
352 views

If $x$ is rational, can $\log(1-x)/\log x$ be algebraic?

If $x$ is positive rational number less than $\frac{1}{2}$, can the following logarithmic expression be equivalent to real algebraic number, say $g$? $$\frac{\log(1-x)}{\log x} = g$$
1
vote
0answers
44 views

Question about the connection between exponential and logarithmic functions

Does this make sense to anyone? What advice would you give me to clarify my reasoning and explanation? One of the really "neat" features of the exponential function: $$f(x)=e^x$$ is the fact that ...
0
votes
1answer
24 views

Is it possible to simplify $y=100x\cdot\log_{x+1} 2$ (Solved)

Is there any way to simplify the following equation, or any way to reconfigure it in a way that is possible to graph? $$y=100x\cdot \log_{x+1} 2$$