Questions related to real and complex logarithms.

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2
votes
4answers
39 views

Logarithmic property justification

I saw this particular line slammed in a proof and it bothers me I can't understand why this is obvious and how would one justify this : $$ 7^{\log (n)} = n^{\log (7)} $$ Can anyone explain ?
0
votes
1answer
19 views

Pull log of a constant out of an integral

Can you pull the log of a constant out of an integral? Can the integral of ln(x/5)•dx become the integral of ln(x)•dx - ln(5) ?
1
vote
2answers
51 views

How to solve $(x-1)e^{-x} > 0.5$

As the title mentioned, how to solve $x$ from the equation: $$(x-1)e^{-x} > 0.5$$ How can I solve this analytically? This is a part of my homework and I got stuck to this equation. I'm also ...
5
votes
3answers
60 views

$\log_{10}(1+10^{-n})<10^{-n}$?

In a paper I was reading, this inequality: $$\log_{10}(1+10^{-n})<10^{-n}$$ came up with no explanation for why it's true. Does anyone have a proof for why this holds? Is there some basic logarithm ...
1
vote
4answers
28 views

Find the derivative of y with respect to the given independent variable

Find the derivative of y with respect to the given independent variable: $y = 3^{-x} \stackrel{D}{\longrightarrow} y' = 3^{-x} \cdot (-1) \cdot \ln 3 $ This is my teacher's solution. I don't ...
0
votes
0answers
23 views

log of summation - what to do when the magnitude of terms is unknown?

I know that ln(a+b) can be rewritten as ln(a) + ln(1+b/a) as long as a>c. Does this last requirement, which I don't really understand, prevent the use of such a tool in working out an equation that ...
14
votes
4answers
111 views

High School Advanced Functions: Clarifying log rules in a log equation - $\log(x^2) = 2$, Solve for x.

I got in an argument with my teacher for the possible solutions of x. From some sources i found that because x is squared, negative values should be possible; however, my teacher insists that: $$ ...
0
votes
1answer
53 views

What's the base of this logarithm?

I'm reading a scientific paper and an equation of the following form appears: x = y log (z). I know what y and z are in my own data set. How do I solve for x? I'm used to logarithms of the form ...
0
votes
1answer
26 views

logarithms laws Question

Say we have $y=e^x$, we then apply to both sides $\ln$ and hence get: $\ln y=x$....apparently from there we get to $y=\ln x$ somehow..although I cannot think of any logarithm law which should yield ...
0
votes
3answers
34 views

How to solve $n$ from $c \leq 1.618^{n+1} -(-0.618)^{n+1}$

I need to solve the bound for $n$ from this inequality: $$c \leq 1.618^{n+1} -(-0.618)^{n+1},$$ where $c$ is some known constant value. How can I solve this? At first I was going to take the ...
2
votes
0answers
39 views

Name of Logarithmic Curve

I was playing with the Desmos graphing calculator and "discovered" the following curve $(\ln x)^2 + (\ln y)^2 = 1$ (I originally found it in the parametric form $(e^{\cos t}, e^{\sin t})$). It would ...
1
vote
2answers
34 views

Find $a$, $b$ and $c$ in $\frac{e^{ax}}{2+bx}=\frac{1}{2}+\frac{x^2}{4}-cx^3$

Find the values of the positive constants $a$, $b$ and $c$ given that when $x$ is sufficiently small for terms in $x^4$, and higher powers of $x$, to be neglected then: $$ ...
2
votes
2answers
44 views

Show that $\ln(1+x)=\ln x+\frac{1}{x}-\frac{1}{2x^2}+\frac{1}{3x^3}-\frac{1}{4x^4}+\cdots$ when $x>1$

If $x>1$ show that $\ln(1+x)=\ln x+\frac{1}{x}-\frac{1}{2x^2}+\frac{1}{3x^3}-\frac{1}{4x^4}+\cdots$ I know from binomial expansion that $(1+x)$ will produce a divergent series in the form of ...
2
votes
1answer
175 views

Logarithmic Series Evaluation

I was trying to generate a direct formula for this series but I am not sure whether it is possible to do so. $$1\ln(1) + 2\ln(2) + 3\ln(3) + 4\ln(4)+\dots+(n-1)\ln(n-1) + n\ln(n)$$
-1
votes
1answer
22 views

big $\Theta$ question dealing with $\log_2{n}$ and $\log_{10}{n}$

Show that $\log_{10}{n} = \Theta(log_2{n})$. I know that I have to show that 1) $\log_{10}{n} = O(\log_2{n})$ show: $\log_{10}{n} \le C * \log_2{n}$ and 2) $\log_2{n} = O(\log_{10}{n})$ show: ...
0
votes
1answer
19 views

Big oh / big theta proof for the following

Find a number $a$ with $s(n) = \Theta(a^n)$ for $s(n) = (\log_2{10})^{(n-3)}$. I'm not quite sure how to proceed. I was having problems with another problem trying to figure out what it means to ...
-1
votes
1answer
27 views

Simplify Logarithmic Equations

How to I simplify these two equations for E 1) 8.6 = 2/3log(E/10^4.4) 2) 5.6 = 2/3log(E/10^4.4)
0
votes
1answer
36 views

Please recognise this natural log calculation algorithm [closed]

x=2^exp*mnt then T=(mnt-SQRT(2))/(mnt+SQRT(2)) ln(x)=( (-2.6398577/(T*T-1.6567626)+1.2920074)*T +0.5 +exp )*LN(2) Can somebody explain this for me? This is from ...
3
votes
1answer
22 views

$y = ln(p+qe^x)/x$, solve $x$

$y = \ln(p+qe^x)/x$ $p$ and $q$ are constants. Express $x$ in terms of $y$. I believe I have to use Lambert W function, but I'm stumped. Thinking help is needed. Thank you very much!
1
vote
0answers
41 views

continuity and limits of $f(x,y)= \begin{cases} \frac{y\ln(x+1)}{y^2+(\ln(x+1))^2} &\text{if $y \neq 0$ }\\0&\text{if $ y=0$}\end{cases}$

Given the set $D:=\{(x,y) \in \mathbb{R}^2: x > -1\}$ and the function $f: D\rightarrow \mathbb{R}$ through $f(x,y)= \begin{cases} \frac{y\ln(x+1)}{y^2+(\ln(x+1))^2} &\text{if $y \neq 0$ ...
0
votes
2answers
32 views

Problem in exponential/log calculus question

I have no idea how to approach this question, $\frac{dQ}{dt} = Q$ and $Q = e$ when $t = 0$, find $Q$ in terms of $t$. I can approach it logically, and the only way $y' = e$ when $t = 0$ is $y= ...
2
votes
1answer
16 views

Rate of decay with half life, present grams and future grams

The half-life of silicon-32 is 710 years. If 80 grams is present now, how much will be present ijn 200 years? I used A(t)=Ae^kt to solve for the rate (k). A(710)=1/2Ae^k(710) 1/2A=Ae^k(710) ...
0
votes
1answer
43 views

Logarithmn subtraction with unknown bases & Logarithmn Identities

The b is supposed to be lowercase in the log functions but I do not know how to do that yet in this syntax. $1)$ Find $x - y$ where $x = 2^{\log_b(3)}$ and $y = 3^{\log_b(2)}$ $2^{\log_b(3)} = ...
1
vote
1answer
67 views

A branch of $\tanh^{-1}z$?

$\def\Log{\operatorname{Log}}$ How can I show that $$\frac{1}{2}\Log\left(\frac{1+z}{1-z}\right)$$ defines a branch of $\tanh^{-1}(z)$ on $\mathbb{C}\backslash((-\infty,-1]\cup[1,\infty))$? (where ...
2
votes
1answer
31 views

Can these rules be used to solve this logarithm?

I saw a video on logarithms saying if there is a limit where $x$ approaches $\pm\infty$ of some fraction, then we can solve by using these rules: If the largest power on the top and bottom are the ...
1
vote
2answers
47 views

Finding the limit $\lim_{n\to\infty} \frac{n\left(\sqrt[n]{n}-1\right)}{\log n}$

I try to calculate the following limit: $$\lim_{n\to\infty}\frac{n\left(\sqrt[n]{n}-1\right)}{\log n}$$ I think it should equal 1, because: $$\exp(x)=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^{n}$$ ...
2
votes
1answer
55 views

Branches of $\log(z)$ on $\mathbb{C}\backslash(-\infty,0]$?

I know this is the most typical example of branches and I think I don't get the concept... Could you help me by giving a detailed development leading to all the required branches? It'd help me ...
3
votes
6answers
63 views

Prove that $\lim_{x \to \infty} \frac{\log(1+e^x)}{x} = 1$

Show that $$\lim_{x \to \infty} \frac{\log(1 + e^x)}{x} = 1$$ How do I prove this? Or how do we get this result? Here $\log$ is the natural logarithm.
1
vote
0answers
16 views

Interpretation of difference log points in a regression

The post "How to interpret the difference in log points" shows how to interpret differences in log values still in log form. As an extension to this, however, I would like to know how to consider an ...
0
votes
1answer
18 views

What is the complexity of halving the size of an $n$-bit number every time.

I was discussing this question with my fiend the other day and was hoping to get some confirmation from someone if the logic I used is correct. Suppose that we have a number $N$ in base 2 ie ...
0
votes
1answer
9 views

Different domains for (apparently) equivalent functions

Let's look at: $f_1(x)=ln(x^2-4)$ $f_2(x)=ln(x-2)+ln(x+2)$ Every high school student can tell they are the same, but the first is defined only for $\{x<-2\}\cup\{x>2\}$, and the latter is ...
0
votes
1answer
17 views

How to deal with such inequalities?

I have that $$ Y \geq n e^{- 1- t \log t + o(1)}$$ and $$Y \leq n e^{\log n +t - t \log t}.$$ Now I would like to find values $t_0(n)$ and $t_1(n)$ such that $$Y \rightarrow 0 \text{ for all } t ...
0
votes
1answer
32 views

Efficient ways to evaluate an integral with a logarithm

Is the approximation in terms of series for the logarithm $$\log(z)= \sum_{n=0}^{\infty}\frac{2}{2n+1}\Bigl(\frac{z-1}{z+1}\Bigr)^{2n+1} $$ a good approximation if I replace this series inside the ...
0
votes
0answers
23 views

Proving an identity from a dilogarithm function. [duplicate]

If $\def\Li{\operatorname{Li}}\Li'_{2}(z) = - \displaystyle\frac{\ln(1-z)}{z}$, how does one get the identity, $$ \Li_{2}\left(- \frac{1}{z}\right) + \Li_{2}(-z) + \displaystyle\frac{1}{2}(\ln(z))^2 ...
6
votes
2answers
38 views

Deriving the analytical properties of the logarithm from an algebraic definition.

Definition: The base $a$ logarithm ($a\in]0,1[\cup]1,+\infty[$) is the continuous function defined by: $\log_a(xy)=\log_a(x)+\log_a(y)~~\forall x,y>0$ and $\log_a(a)=1$ If I used this definition ...
6
votes
3answers
109 views

How to calculate $\lim_{x \to0} \dfrac{f(x)-f(\ln(1+x))}{x^{3}}$

$f$ is a differntiable function on $[-1,1]$ and doubly differentiable on $x=0$ and $f^{'}(0)=0,f^{"}(0)=4$. How to calculate $$\lim_{x \to0} \dfrac{f(x)-f\big(\ln(1+x)\big)}{x^{3}}. $$ I have ...
6
votes
5answers
535 views

How do you solve a logarithm with a non-integer base?

How would one calculate the log of a number where the base isn't an integer (in particular, an irrational number)? For example: $$0.5^x = 8 \textrm{ (where } x = -3\textrm{)}$$ $$\log_{0.5}8 = -3$$ ...
1
vote
1answer
61 views

Show that $\log \log z$ is analytic

Show that $Log( Log z$) is analytic in the domain consisting of the $z$ plane with a branch cut along the line $y = 0, x ≤ 1$. As of now im not too sure on how to solve this problem, so i was ...
0
votes
1answer
21 views

How do I solve $0 = x\times114 - x\times\log_3(x) - 20.28\times y$ in matlab for different values of $y$?

I have $y = 10^3, 10^6, 10^9, 10^{12}, 10^{15}, ...$ and above mentioned equation. How do I solve (i.e. getting values of x for different y) and plot this equation in MATLAB ?
2
votes
3answers
81 views

Solving $\ln(x) = e^{-x}$

I'm trying to solve $\ln(x) = e^{-x}$ but I can't really get how to do it :( (Removing a statement that was incorrect, as explained by the comments below) Additionally, while I started to solve it I ...
0
votes
0answers
11 views

maximization of a function with random variable

I would like to know whether this is true in general, and if not when this can be. I am not sure and so I am mostly asking for confirmation. So, is the following correct ? $$\log [\max_{x} ...
1
vote
2answers
44 views

Difference between `log n` and `log^2 n`

I'm researching the different execution time of various sorting algorithms and I've come across two with similar times, but I'm not sure if they are the same. Is there a difference between ...
2
votes
4answers
39 views

Solving inequality with logarithms.

I was playing around and found this $$x\log a\le a-1$$ Solve for $x$ in the above equation, where $a>0$ My attempt $$\log a\le \frac ax-\frac1x$$ $$a\le \frac{\exp(a/x)}{\exp(1/x)}$$ But I ...
0
votes
1answer
38 views

Find all odd numbers $n$ such that $q= \frac{\ln(3n+1)}{6\ln(2)}$ is also an odd number. [closed]

Find all odd numbers $n$ such that $$ q= \dfrac{\ln(3n+1)}{6\ln(2)}$$ is also an odd number.
1
vote
2answers
52 views

Why can we take the log of both sides?

I was watching a video that proves the "Log of a power" rule. I'm just having trouble understanding the loga(a^x) = x rule - which he uses in the proof And I don't get why you can log both sides. ...
3
votes
1answer
34 views

Find derivative of tricky logarithmic functions

Find the derivative of $y=(x^{x+1})(x+1)^x$ So this is what I have, $$\ln y=\ln[(x^{x+1})(x+1)^x]$$ $$= \ln x^{x+1} + \ln(x+1)^x$$ $$\frac{1}{y}y' = (1)(\ln x) + (x+1)\frac{1}{x} + (1)(\ln(x+1)) + ...
0
votes
2answers
38 views

For which values of $m$, $f(x)=mx$ intersect the function $g(x)=\log x$?

For which values of $m$, the function, $f(x)=mx$ intersect the function, $g(x)=\log x$ I suppose that this problems reduce to the next form. Find for which values of m, exist solution for the ...
1
vote
0answers
41 views

Sum of 2 different irrational logarithms = Irrational?

I am having some problems proving that the following sum is irrational or rational: $\log_2(3)+\log_3(2)$ = irrational. This is all I've got for now: $\log_2(3)=\frac mn \iff 2^{\frac mn}=3 \iff ...
1
vote
0answers
16 views

Logarithm of an applied permutation

Say I have a cyclic permutation $P$, a known input $x$, and a known output $y$ such that $$y = P^a x$$ for some $a$. Is there a good way to search for $a$ (i.e. better than brute force)? Are some ...
2
votes
5answers
192 views

Textbook clarification: $\log = \ln$

Textbook reads: All logarithms are natural logarithms: $\log = \ln$. Does this mean $n\log(n) = n\ln(n)$?