Questions related to real and complex logarithms.

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3
votes
3answers
85 views

Logarithmic Equation: Solve for $x$

$$\log_{3x}81 = 2$$ How would I go about solving this? This is what I tried: $$\log_{3x}81 = 2$$ $$\frac{\log81}{\log 3 + \log x }= 2$$ Where do I go from here? If I isolate $\log x$ on one side, ...
2
votes
2answers
58 views

Solve the following exponential equation

$$7^{3x+1}=5^x$$ I am trying to solve this equation. I solved the equation and got what I believe to be the correct answer, but when I verify the answer it appears to be incorrect. Any idea why? Here ...
0
votes
1answer
56 views

Logarithm as a limit of a decreasing sequence

Let $k : \mathbb{N}^{>0} \to \mathbb{R}$ be such that $k_n = n(t^{1/n} - 1)$, where $t \in [0, 1] \subset \mathbb{R}$. Note that $\lim_{n \to \infty} k_n = \log(t)$. The plot of $k$ for a given $t$ ...
3
votes
0answers
17 views

limit of a sequence of iterated logarithms [duplicate]

I was playing around with the family of sequences $s(x)$ defined for $x > 0$ as $s(x)_0 = x$, $s(x)_{n+1} = \log(1+s(x)_n)$ and I noticed a strange behavior. Specifically, regardless of the choice ...
-2
votes
1answer
51 views

Is this identity correct?

Is this identity true? Wolfram|Alpha thinks is not. $$x^{ln(x^3)} = e^{3\,[ln(x)]^2}$$ That's how I demonstrated it: $${\left(e^{ln(x)}\right)}^{3\,ln(x)} = e^{3\,[ln(x)]^2}$$ ...
1
vote
1answer
55 views

Induction of factorial

I was perusing the wikipedia page on Mathematical induction, and it mentions it's possible to prove by induction that. $\frac{n^{n}}{3^{n}}<n!<\frac{n^{n}}{2^{n}}$ for $n\geq6$ Proof for $n=6$ ...
1
vote
1answer
52 views

Proof of discrete logarithm?

If you have that $a$ is a primitive root mod p. How can you prove this discrete logarithm property? $log_{a}(b_1b_2) = log_{a}(b_1) + log_{a}(b_2)$ (mod $p-1$) I see the proof for the regular ...
7
votes
3answers
146 views

Finding $\int_0^{\pi/4}\sqrt{1+\left( \tan x\right)^2}dx$

I would like to understand all the steps to find out this integral $$ \int_0^{\pi/4} \sqrt{1+\left( \tan x\right)^2} dx$$ Wolfram Alpha returns: $$ \frac12 \log(3+2 \sqrt2) = 0.881373587019543...$$ ...
3
votes
2answers
50 views

Calculate the following integral: $\int \sqrt[3]{1+x\ln{x}} \cdot (1+\ln{x}) dx$

I have to calculate the following integral $$ \int \sqrt[3]{1+x\ln{x}} * (1+\ln{x}) dx$$ I have thought about using the following notation: $$ t = {1+x\ln(x)} \Rightarrow x\ln{x} = t-1 $$ But ...
1
vote
1answer
35 views

Simplify $log(sinhz)$ when $|z|$ tends to $0$?

I was given $\log(\sinh z)$ and I need to show it tends to $\log z$ when $|z|$ tends to $0$. I have tried converting $z$ to $x+iy$ then split $\sinh z$ but that doesn't seem to get me anywhere. I ...
0
votes
1answer
55 views

The limit of $ m\int_{a}^{1/m} \frac{dx}{x}=0 $ and $ m\int_{a}^{\infty} \frac{dx}{x^{1+m}}=0$ as $m\to0$

Given $ a >0 $ is it correct that $$ \lim_{m\to 0}m\int_{a}^{1/m} \frac{dx}{x}=0 $$ by the properties of the logarithm function? Or on the other hand, $$\lim_{m\to 0} m\int_{a}^{\infty} ...
0
votes
2answers
35 views

Adjusting the steepness of a curve

I've got an array of numbers, each between 1 and 0, sorted in descending order. When i put those numbers on a graph, they decrease too quickly: I would like to create a function ...
1
vote
1answer
26 views

prove logarithmic inequality for N>1200

For N > 1200 how can i prove that 3.09N/Log(N) + 1 <= 1.7(2N+1)/Log(2N+1) (sorry, could not figure out how to put the 'less than or equal' symbol there, tried \leq)
1
vote
3answers
55 views

Is this power rule true for the natural base?

Two questions 1) I was wondering if $e^{k \ln{x}}=k$ for any k. Is it? 2)To test I went to Maple and typed e^-ln(x) and it gave $e^{-ln(x)}$. I tried simplify and ...
4
votes
4answers
994 views

How is $\ln(-1) = i\pi$?

How do I derive: $\ln(-1)=i\pi$ and $\ln(-x)=\ln(x)+i\pi$ for $x>0$ and $x \in\mathbb R$ Thanks for any and all help!
2
votes
4answers
52 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
144 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
60 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
62 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
312 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
61 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
180 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
106 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
32 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
37 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
53 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
37 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
54 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
197 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
27 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
29 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
38 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)
3
votes
1answer
30 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
54 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
38 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
110 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
48 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
138 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
45 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 ...
4
votes
3answers
202 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: $$\exp(x)=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^{n}$$ ...
3
votes
1answer
122 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
71 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
265 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
30 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
25 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
22 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
36 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
28 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
56 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
113 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 ...