Questions related to real and complex logarithms.

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0
votes
1answer
240 views

Converting base2 scientific notation to base10 scientific notation

I would now like to know how to convert a value in base2 scientific notation (is that the correct terminology?), say 1.93 * 2 ^ 88, into the form of A * 10 ^ B. I want to do this without expressing ...
2
votes
2answers
73 views

Find $n$ satisfying the equation $[\log_21]+[\log_22]+[\log_23]+\dots[\log_2n]=1538 $

If $[\cdot]$ denotes greatest integer function, then what is the value of natural number $n$ satisfying the equation $$[\log_21]+[\log_22]+[\log_23]+\dots[\log_2n]=1538 ?$$ My try: Note that ...
3
votes
2answers
103 views

Apply the natural logarithm fractional number of times

Let f_n(x) be the recursive function that adds 1 to x and takes the natural logarithm, ...
0
votes
2answers
115 views

How to prove $\sum\limits_{k=2}^{n}\dfrac{1}{k}<\log(n)<\sum\limits_{k=1}^{n-1}\dfrac{1}{k}$

How to prove $\sum\limits_{k=2}^{n}\dfrac{1}{k}<\log(n)<\sum\limits_{k=1}^{n-1}\dfrac{1}{k}$ It is clear if i consider the area under $f(x)=\dfrac{1}{x})$ from $1$ to $n$ end divide the ...
0
votes
1answer
27 views

Show that the following holds;

Let $h(p) = -p \log p-(1-p)\log (1-p)$ denote the binary entropy of a Bernoulli distribution when the probability of observing a zero is $p$, where $\log$ denotes the logarithm to base 2. Show, using ...
0
votes
2answers
51 views

proof of logarithmic property $\displaystyle a^{\log_{a}{b}}=b$

I don't know how to show that $\displaystyle a^{\log_{a}{b}}=b$ Can anyone give a hint?
35
votes
5answers
706 views

How to find $\int_0^\infty\frac{\log\left(\frac{1+x^{4+\sqrt{15}}}{1+x^{2+\sqrt{3}}}\right)}{\left(1+x^2\right)\log x}\mathrm dx$

I was challenged to prove this identity $$\int_0^\infty\frac{\log\left(\frac{1+x^{4+\sqrt{15\vphantom{\large A}}}}{1+x^{2+\sqrt{3\vphantom{\large A}}}}\right)}{\left(1+x^2\right)\log x}\mathrm ...
0
votes
1answer
18 views

Creating a constrained log function

Good morning, I have a series of values that I intend to use as the exponents and I would like to create a log function so that: $Log_x(y_1)=.1$ $Log_x(y_2)=.2$ $Log_x(y_3)=.3$ ... ...
1
vote
1answer
43 views

How to prove this Logarithmic identity?

I am just learning logs, and can't get this one out ? How to prove that $$ \log_a (b) \cdot \log_b (c) = \log_a (c) $$ in the format : log [base]([argument]) ...
1
vote
2answers
54 views

Logarithmic Differentiation

When do we use : $ \ln(ab) = \ln a + \ln b $ and when do we use : $ \ln |y| = \ln |f_1(x)| + \ln |f_2(x)| + \cdots + \ln |f_n(x)| $ ? It is stated that we use the second form of log differentiation ...
5
votes
2answers
65 views

How to find the limit of $\lim_{n\to \infty} n(H(n) - \ln(n) - \gamma)$

How to find the following limit: $$\lim_{n\to \infty} n(H(n) - \ln(n) - \gamma)$$ where $H(n) = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$ is the $n^{th}$ harmonic number and $\gamma$ is the Euler ...
0
votes
1answer
51 views

What is the same as the inverse of a logarithm?

I am trying to simplify $f(n) = \frac{n}{\log(n)}$ into a more easily understandable function. Up until now, I got as far as $n\cdot(\left(\log(n)\right)^{-1})$. Is there any way I can further ...
0
votes
2answers
88 views

Solve equation with logarithm base 10

I am going back to study log and unfortunately I don't know a lot. I need to solve this: $$ 100= 10\log_{10} \left(50/x\right) $$ I did the wrong calculation just moving stuff to the left, but ...
8
votes
4answers
3k views

Approximating Logs and Antilogs by hand

I have read through questions like Calculate logarithms by hand and and a section of the Feynman Lecture series which talks about calculation of logarithms. I have recognized neither of them useful ...
0
votes
3answers
31 views

Compare two powers

how can I compare these powers: $3^{500}$ and $5^{300}$ What I did is: $\log_3(3^{500})$ and $\log_3(5^{300})$ So I have $500$ and $\log_3(5^{300})$ Now I do not know what to do. Thank you in ...
2
votes
1answer
74 views

How to solve $6^{2x}-10\cdot 6^x=-21$ using logarithms?

What do I do with $\large 6^{2x}-10\cdot 6^x=-21$? Since $6$ and $-60$ are not of the same base (nor can they be written as exponents of the same base cleanly) I am having trouble solving for ...
1
vote
2answers
53 views

Help me prove the result of this limit: $\lim_{x\to \infty} {\log_{x^3+1} (x^2+1) \over \log_{2e^x+1} (x^2+5x)}$

The limit is $$\lim_{x\to \infty} {\log_{x^3+1} (x^2+1) \over \log_{2e^x+1} (x^2+5x)}$$ I know that it should be equal to $\infty$ but i have yet to prove it. Please help me do so.
2
votes
1answer
41 views

Definite integral of partial fractions?

So I'm to find the definite integral of a function which I'm to convert into partial fractions. $$\int_0^1 \frac{2}{2x^2+3x+1}\,dx$$ Converting to partial fractions I get... $\frac{A}{2x+1} + ...
1
vote
1answer
68 views

Derivative of $\operatorname{Log}(\operatorname{Log}(z^2))$

Please help me with this question: (i don't know how to start) Suppose that $f(z)$ = $\operatorname{Log}(\operatorname{Log}(z^2))$. Find $f'(z)$ where it exists, and determine the set of points at ...
7
votes
1answer
152 views

Differentiating $y=x^x$ with the formal definition of a derivative

A friend and I were messing around with derivatives, and while we both know the procedure for finding the derivative of $y=x^x$ with logarithmic differentiation, i.e. $$y=x^x\\ ln(y)=x*ln(x)\\ ...
0
votes
1answer
43 views

How solve this $\log { { x }^{ \log _{ x }{ y } } } =\quad \frac { 5 }{ 2 } \\ x+y=6\\ $

How solve this logarithm equation $\log { { x }^{ \log _{ x }{ y } } } =\quad \frac { 5 }{ 2 } \\ x+y=6\\ $
0
votes
1answer
206 views

Give the domain and range of $y=\log(x-3)+2$

I am so confused. I think the domain is $x>3$ but is the range ARN or is it $y>0$ . . .
3
votes
1answer
101 views

How to solve $x^{\log_3(x)} \geq \frac{1}{27}$

How to solve this? My problem is to solve: $$x^{\log_3(x-4)} \ge \frac{1}{27}.$$ The log base is $3$.
1
vote
1answer
71 views

How solve this logarithms equation

What relationship between a,b and c ?
3
votes
1answer
65 views

Logarithms, prove this limit.

Mathematica knows that: $$\log (n)=\lim_{s\to 1} \, \left(1-\frac{1}{n^{s-1}}\right) \zeta (s)$$ Kind of tautological starting with logarithms, but I would like to know better why this limit works: ...
1
vote
3answers
42 views

How to move from powers to simple logarithms?

I'm following a book that briefly moves from $$16000 \times 2^{\displaystyle \left (-\frac{x}{24} \right )} = 1600$$ to $$x = \frac{24 (\log(2) + \log(5))}{\log(2)}$$ adding the comments that ...
6
votes
1answer
135 views

How to compute the mean average exponent of the naturals? What is the limit for large numbers?

With a friend I was trying to get an understanding for why the expected gap between primes is logarithmic. With that motivation I tried to express the average exponent of numbers. By average ...
0
votes
1answer
131 views

Linear to semi-logarithmic scale

I've got some FFT results I want to draw with a log10 scale on the x axis. Let's call nBins the number of bins (window size / 2) nPixels the total number of pixels We will assume that the ...
1
vote
2answers
35 views

How is natural log integration broken up into this range? (equation is contained the script)

When I was reading a paper, I found an strange derivation like $$\int^{+\infty}_{-\infty}\mathrm{ln}(1+e^w)f(w)dw\\=\int^0_{-\infty}\ln(1+e^w)f(w)+\int^\infty_0[\ln(1+e^{-w})+w]f(w)dw$$ when $w$ is ...
0
votes
2answers
82 views

Evaluate $\frac{d}{dx}\{(\sin x)^{\cos x} + (\cos x) ^{\sin x}\}$ with logarithmic differentiation

Spivak asks us to evaluate $$\dfrac{d}{dx}\{(\sin x)^{\cos x} + (\cos x) ^{\sin x}\}$$ by logarithmic differentiation. Does he mean for us to evaluate each term separately (which seems to turn out to ...
4
votes
6answers
114 views

$\lim_{n\to\infty}\left(1+\frac{3}{n}\right)^\frac{n}{2}$

I am trying to resolve this to number $e$. However, I would like to do it in the simplest form. just a note I already tried wolfram but I would like someone to give me a simpler solution. ...
-2
votes
2answers
139 views

Prove that $e ^ π$ > $π ^ e$. [duplicate]

Prove that: $$e ^ π > π ^ e.$$ Hint: Take the natural log of both sides and try to define a suitable function that has the essential properties that yields the above inequality
12
votes
2answers
341 views

$\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$

Prove that: $\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$, for $n > 1,\, n\in \mathbb{N}$ For example. For $n=2$, we have $\lfloor ...
0
votes
0answers
40 views

Why can't the base of a logarithm be negative? [duplicate]

I understand why the base of a logarithm can't be 0 or 1, but why negative? What I found out is that when the base is negative we get imaginary results when the powers are rational numbers with odd ...
8
votes
3answers
12k views

Why must the base of a logarithm be a positive real number not equal to 1?

Why must the base of a logarithm be a positive real number not equal to 1? and why must $x$ be positive? Thanks.
0
votes
1answer
50 views

Expected Value of Exponential

I want to calculate $\log E[\exp(-\sqrt{d} S \epsilon)]$, where $\epsilon \sim N(0,1)$ and everything else is deterministic. The result should be $\frac{d}{2}||S||^2$ but why?
0
votes
1answer
47 views

Prove $\displaystyle n\log\left(1+\frac{1}{n}\right) - \log\left(1+\frac{1}{n+1}\right) < \log\left(1+\frac{1}{n+1}\right)$

I'm trying to prove the above inequality, assuming $n\ge1$. I've been working on this one using log properties and trying to reduce this inequalitiy to simpler ones. Though!, is it even correct? or am ...
0
votes
1answer
65 views

Numerical Analysis - show something about the rate of convergence

We are given an iterative method for finding roots, $x_{n+1}=g(x_n)$, we are given the rate of convergence of this method is $p$, and also that: $$\lim _{n \to \infty} \frac{|e_{n+1}|}{|e_{n}|^p} = ...
2
votes
1answer
621 views

Find the volume of the solid obtained by rotating the region bounded by $y = ln x$, $y = 0$, $x = 2$ about the $x$-axis

I have the problem: Assuming $y = ln(x)$, and $y = 0$, find the volume bound by these two lines and the point $x = 2$ if the area were rotated around the $x$-axis. I ended up with $2\pi\int_1^2 ...
4
votes
3answers
110 views

Difficult Integral Involving the $\ln$ function

Please help me solve this integral! I have tried multiple different procedures for integration by parts, as well as substitution and have not come up with anything. $$\int\frac{\ln x}{(\ln x+1)^2}dx$$ ...
3
votes
1answer
37 views

What is the number of real roots of $(\log x)^2- \lfloor\log x\rfloor-2=0$ $\lfloor\,\cdot\,\rfloor$ represents the greatest integer

Question : What is the number of real roots of $(\log x)^2- \lfloor\log x\rfloor-2=0$. $\lfloor\,\cdot\,\rfloor$ represents the greatest integer function less than or equal to $x$. I know how to ...
2
votes
2answers
29 views

Let $x_0 > x_1 > x_2>x_3$ be any positive real numbers . What is the largest value of the real number k such that..

Question : Let $x_0 > x_1 > x_2>x_3$ be any positive real numbers . What is the largest value of the real number k such that $$\log \frac{x_0}{x_1}1993 + \log \frac{x_1}{x_2}1993 +\log ...
1
vote
1answer
51 views

Comparing logarithmic functions. Master Method

I'm learning the master method and am looking for help on how to best approach comparing two functions asymptotically. More specifically, I have: ...
0
votes
1answer
48 views

show that $(1+ \frac {x}{n})^n < e^x$ and $e^x < (1- \frac{x}{n})^{-n}$ if $x<n$

If $n$ is a positive integer and if $x>0$,show that $(1+ \frac {x}{n})^n < e^x \quad$ and that $\quad e^x < (1- \frac{x}{n})^{-n} \quad $ if $x<n$ I proved the first one by the ...
1
vote
2answers
64 views

Solving equations with logarithms

I'm having trouble with solving equations that has logarithms in them. For example: $$x^{\log(x)} = \frac{100}{x}$$ How can I solve this? I have reed about how to do it but when I try to do the same ...
1
vote
1answer
35 views

Differentiating logarithms

I am trying to prove that $$ f(x) = ^alog(x) => f'(x) = \frac {1} {ln(a)*x} $$ So I start at$$ f(x) = ^alog(x) $$ Then I move to:$$ f(x) =\frac {ln(x)} {ln(a)} $$ And there I get stuck: I want ...
1
vote
1answer
151 views

How do computers calculate the log of a value? [duplicate]

I'm not sure if this question belongs on StackOverflow or here (please let me know if the former, and i'll delete this and ask there), but I was wondering how the ...
-1
votes
1answer
48 views

Rewrite a formula in terms of exponential to the power of logarithm

I would like to rewrite the following formula, f(x). how can I rewrite the f(x) $$ f(x) = ...
2
votes
1answer
177 views

Logarithms melting my brain

So I've got an inequality: $\ln(2x-5) > \ln(7-2x)$ and I attempt to solve by doing the following: $$\frac{\ln(2x)}{\ln(5)} > \frac{\ln(7)}{\ln(2x)}$$ $$\Rightarrow \ln(2x) \cdot \ln(2x) > ...
14
votes
2answers
1k views

Show that these two numbers have the same number of digits

I want to show that for $n>0$, $2^n$ and $2^n + 1$ have the same number of digits. What I did was I found that the formula for the number of digits of a number $x$ is $\left ...