Questions related to real and complex logarithms.

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0
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1answer
41 views

How to simplify this ln?

I am solving this problem $$\int {\frac{1}{\sqrt{9x^{2}-25}}dx}$$ the step I got stuck, $$\frac13 \ln{\left(\frac{3x+\sqrt{9x^2-25}}{3}\right)} $$ and in my textbook the answer was $\frac13 (\ln{{(...
15
votes
3answers
2k views

Why is Euler's number used as a base for logarithms? [duplicate]

Is there some special property of '$e$' which makes it suitable to be used as a base for logarithms? Moreover, does the natural logarithm possess some advantage over the common logarithm? I don't ...
2
votes
1answer
18 views

How to find the highest [natural] radix base of a given number with a natural output

Like the title says, I'm trying to make a program that finds the highest natural radix of a given number with a natural output. My program works, but it loops every number possible number up to a ...
2
votes
2answers
34 views

Prove $\log u > \frac{u - 1}{u}$ for $u > 1$

How to prove that for $u > 1$ $$\log u > \frac{u - 1}{u}$$ without using integrals? I think I'm supposed to use derivatives or Taylor's theorem, as the exercise comes from a lecture about these ...
1
vote
1answer
25 views

Omitting the refference to a particular logarithmic base - order notation

How can I prove by using the Order notation definition that we can conventionally refer to an algorithm taking "log time", without referring to a particular logarithmic base?
0
votes
1answer
74 views

How is this logarithm problem solved?

If $\dfrac{xy\log xy}{x+y} = \dfrac{yz\log yz}{y+z} = \dfrac{zx\log zx}{z+x}$ show that $x^x = y^y = z^z$. I tried equating to a constant $k$ and adding up I also tried adding up the numerators and ...
2
votes
1answer
39 views

Find the possible values of $x$ if $2^{2x+1} = 3(2^x) -1$

Find the possible values of $x$ if $2^{2x+1} = 3(2^x) -1$ I know that $x=0$ and $x=-1$ are possible values of $x$ by looking at the equation. I need help understanding how to use logarithms to ...
0
votes
1answer
49 views

Differentiating a Log Function [closed]

My current $\ln$ function is: $$ \ln\Big(\frac{y}{x}\Big) = b_1 + b_2 ln(x). $$ How would I find the elasticity for this model? What I have done so far is convert the equation to: $$ \Big(\frac{y}{x}...
2
votes
1answer
47 views

What's wrong with this wrong derivation of $\ln(-x)$?

If we take $\ln(x) = \int_{1}^{x} \frac{1}{y} dy$ as the definition of logarithm, then I am ending with a stupid derivation of $\ln(-x)= - \int_{-x}^{-1} \frac{1}{y} dy$. From the definition, if I go ...
3
votes
1answer
36 views

Density of $\{ \ \{\ln k\} \ \}_{k=1}^{\infty}$

Is this sequence dense in $(0,1)$? I want to say that it is, I think the transcendence of the logarithm function is leading me to believe that it is, but I don't know how to prove it.
0
votes
0answers
16 views

logarithm of odds score at test value of 0

In my human genetics class, we learned to compute a logarithm of the odds (LOD) score as follows: $LOD=\log_{10}{\frac{\theta^r(1-\theta)^{n-r}}{0.5^n} }$, where $\theta$ is a test value for ...
1
vote
2answers
36 views

Calculating the discrete logarithm

I'm given a prime number $p = 1217$ I'm also given the following equations: $ 40 = \log2 \mod 64 $ $ 63 = \log3 \mod 64 $ $ 13 = \log5 \mod 64 $ $ 13 = \log2 \mod 19 $ $ 10 = \log3 \mod 19 $ $ ...
0
votes
2answers
28 views

How do I solve for x? Do I need the Lambert W function?

I need to solve the next equation x: $d-x+yln[\frac{d}{x}]=b$ y, d, b, and x are all real, positive numbers. How do I solve for x? Do use the lambert W function and if so how is that done? Thanks!
0
votes
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
votes
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 ...
0
votes
0answers
16 views

Find the complex potential given the real part $\frac{6000}{\pi}\theta$

In my 'Complex Variable Theory' notes, we have an example where we use a linear fractional mapping to solve for Laplace's equation between semi-circular plates. The Question reads: Find the ...
1
vote
1answer
35 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 ...
0
votes
1answer
41 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 ...
1
vote
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
vote
0answers
13 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
vote
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.
1
vote
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
27 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( \sum^n_{i=1}x_i^{\...
1
vote
0answers
79 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= \int^1_0\...
4
votes
1answer
99 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
41 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
votes
1answer
34 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
votes
1answer
25 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
vote
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
vote
2answers
46 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
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 ...
-1
votes
2answers
45 views
0
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3answers
40 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
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
vote
1answer
14 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
vote
2answers
60 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
vote
1answer
32 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
89 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.
1
vote
0answers
30 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 (\exp[1/...
12
votes
2answers
212 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
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
0answers
21 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
28 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?