Questions related to real and complex logarithms.

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2
votes
1answer
24 views

Is there any way to extend the domain of this function through analytic continuation?

$\prod_{k=2}^x \log k=F(x)$ It looks a lot like the gamma function (a sort of logarithmic factorial), and I wonder if it can be similarly expressed as an integral or something. Any ideas?
4
votes
4answers
273 views

The caret ^ symbol means exponentiation informally in math. Why not a symbol for log too? [closed]

Plus, minus, multiply, divide, and exponentiation all have symbols in math (+, -, *, /, ^ ) . But why isn't there the missing log symbol too? Here's how it would work: 4 ^ 5 = 1024 (as is standard ...
12
votes
5answers
2k views

Understanding imaginary exponents

Greetings! I am trying to understand what it means to have an imaginary number in an exponent. What does $x^{i}$ where $x$ is real mean? I've read a few pages on this issue, and they all seem to ...
0
votes
1answer
96 views

The value of the logarithmic expression can never be $\ldots$

The value of the logarithmic expression $\log_x \dfrac{x}{y} +\log_y \dfrac{y}{x},\text{where}\quad x\geq y>1\quad$ can never be $\bf\text{options}$ a.) $-1\quad$ b.)$\quad0.5\quad$ c.) ...
0
votes
1answer
16 views

How to solve this Diophantine equation (involving natural logarithms)?

The equation is $r = \ln{a} + b \ln{c}$ where $r \in \mathbb{R}$ is fixed and $a,b,c \in \mathbb{N}$. In other words, for arbitrary real r, how can one say whether a solution (in form above) exists ...
1
vote
2answers
38 views

How can I isolate for the $z$ exponent?

Can anyone help me with this math equation? Solve for $z$ $$P = \frac{e^z}{1 + e^z}$$ $$P(1 + e^z) = e^z$$ $$P + Pe^z = e^z$$ $$P = e^z - Pe^z$$ I've got this far, am I at least on the ...
15
votes
2answers
44k views

How to type logarithms in Wolfram|Alpha?

Its sometimes hard to type it if logarithm is not natural and base is not 10, especially if base is variable. So anyone know rules how to type?
0
votes
1answer
32 views

Need help with logarithmic differentiation

I need to use logarithmic differentiation to get f(x)=x$\sqrt{(x+1)(x+2)(x+3)(x+4)}$. I've been working on it for a while and could use some help. Thanks!
0
votes
2answers
75 views

Solve $2^x=13 \bmod 3^4$

Solve $2^x=13\bmod 3^4$ I know $\log13=30\bmod 3^4$ and $\log16=15 \bmod 3^4 $ I've tried subbing $\log13/\log16$ for $2$ but I am not sure what to do next.
0
votes
0answers
27 views

Are Percentile and Logarithm Exchangable?

I am wondering if the Percentile Operator, denoted as 'prctile', commutates with the logarithm of basis 10, denoted as 'log10'. Is the following statement true, and if yes, why? log10(prctile(X)) = ...
2
votes
1answer
60 views

How to solve this logarithmic inequality?

I've started a data structure course and I need some help with solving these logarithmic inequalities. It would also be helpful because later on these kind of calculation won't pose a problem later ...
0
votes
0answers
24 views

Need help to Compute a specific derivation

I have a function equal to: Where L is an orthonormal matrix of eigenvectors of a matrix S, and ...
5
votes
3answers
8k 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.
1
vote
2answers
52 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 ...
1
vote
1answer
43 views

Could someone explain steps?

I am learining about logarithm equations, and i can´t seem to understand how to solve such an equation, could someone help? I must solve the equation/find $x$ for: $$2^{2x} - 3\cdot2^x - 10=0$$ The ...
0
votes
1answer
49 views

Logarithm inequality for specific range

I need to show that: $$ \ln(1+x)\left(\ln\left(\frac{1+x}{1-x}\right)+1\right)+\ln(1-x)\ge 0, $$ for $0\le x\le 2/3$. Thanks
1
vote
3answers
39 views

Question regarding logarithms 2

What is $\ln(-1)$? And would there a taylor series for $$\ln\frac{1+x^m}{1-x^m}$$?
1
vote
2answers
60 views

Find $\lim\limits_{n \to \infty} \frac{\log(1+2^n)}{\log(1+3^n)}$

How to calculate this limit? $$\lim\limits_{n \to \infty} \frac{\log(1+2^n)}{\log(1+3^n)}$$
0
votes
1answer
12 views

Solve for r. Logarithms

$$ 36000 = 3450 * \frac{1-[1/(1+r)^{12}]}{r} $$ The next step is divide both sides by 3450. Now I'm stuck. Help solve for r.
22
votes
1answer
446 views

The positive root of the transcendental equation $\ln x-\sqrt{x-1}+1=0$

I numerically solved the transcendental equation $$\ln x-\sqrt{x-1}+1=0$$ and obtained an approximate value of its positive real root $$x \approx 14.498719188878466465738532142574796767250306535...$$ ...
-1
votes
1answer
42 views

Why is $\frac{\sum_{i=1}^n \log(X_i)}{n} = \overline{log X}$ [closed]

Why is $$\frac{\sum_{i=1}^n \log(X_i)}{n} = \overline{log X}$$ ($X_i$ are i.i.d samples)
0
votes
3answers
38 views

Question regarding logarithms

Can you factor out the $m$ out of $\ln(c\cdot x^m)$ where $c$ is a constant?
1
vote
2answers
56 views

Why does the log-normal probability density function have that extra “x”?

For a random variable $X \sim N(\mu, \sigma^2)$, the probability density function is $$f(x; \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} \cdot \exp\left\{ -\frac{(x-\mu)^2}{2\sigma^2} \right\}$$ ...
1
vote
0answers
13 views

Calculating gain ratio from a dB value

In a practice problem I have: power gain = $\log_{10}(\frac{db}{20})$ The final answer for the ratio is 1. The dB value is $-3$. When I do $\log_{10}(\frac{3}{20})$ I get $-0.823$. Just wondering ...
0
votes
1answer
44 views

How do we solve for $n$?

Asymptotic complexity gives an idea of how rapidly the space/time requirements grow as problem size increases. • Suppose we have a computing device that can execute 1000 complex operations per ...
5
votes
4answers
263 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 ...
19
votes
3answers
706 views

Evaluating $\int_0^\pi\arctan\bigl(\frac{\ln\sin x}{x}\bigr)\mathrm{d}x$

I found the following integral as a by product of another one. It has a nice closed form. $$ \int_{0}^{\pi} \arctan\left(\ln\left(\sin x \right) \over x\right)\,{\rm d}x $$ Mathematica and ...
0
votes
2answers
851 views

Express the logarithm in terms a,b,c

Suppose that: $\log_{10}A = a$ $\log_{10}B = b$ $\log_{10}C = c$ I need to express the following in terms of $a$,$b$,$c$. $\log_{10}A + 2\log_{10}(1/A)$ $\log_{10}(((AB)^5)/C)$ ...
4
votes
3answers
81 views

Closed form for the partial sum $\sum\limits_{k = 1}^n \frac{\ln k}k$

I'd like to find a closed form for this partial sum: $$\sum\limits_{k = 1}^n \frac{\ln k}k$$ Using the properties of the logarithms, I converted the above into $$\ln\left(\prod_{k = 1}^n ...
8
votes
3answers
256 views

Is $\ln(x)$ ever greater than $x$?

Is $\forall x \in \mathbb{R}, \ln(x) \lt x$ a true statement? Just wondering for some convergence related thing
-2
votes
4answers
64 views

Logarithm and trigonometry

Is $\ln (\sin x-\cos x)$ equal to $\ln (\cos x-\sin x)$? So I did a integral problem but the answer is not same the answer given. I'm given this question $\int (\frac{2}{1-\tan x})dx$ So I got ...
0
votes
1answer
33 views

Iterative Logarithm in Recurrence Relation?

Anyone Could describe me How we can solve this recurrence relation? $T(n) = T(\log n) + O(1)$ $T(1) = 1$ a) $O(\log n)$ b) $ O (\log^* n) $ c) $ O (\log^2 n) $ d) $ O (n / \log n) $ Our TA ...
1
vote
0answers
24 views

Minimum of the difference of two logarithms

I am trying to find an analytical expression of the minimum of $$ f_n(x) = \frac{2x}{n^2+n}\log(x) - \frac{2x+2}{n^2+3n+2}\log(x+1) $$ when $x\in [1;n]$ I used to think from graphing it that this ...
2
votes
1answer
57 views

Show $\log(1+x)$ is not a contraction mapping

Show $F:[0,\infty] \to [0,\infty]$, $F(x) = \log(1+x)$ is not a contraction mapping. Attempt: Assume $F$ is a contraction mapping, then we have that $\forall x,y \in [0,\infty)$, $|F(x) - F(y) | ...
2
votes
4answers
2k views

How can I solve for $n$ in the equation $n \log n = C$?

Believe it or not, this isn't homework. It's been many years since grade school, and I'm trying to brush up on these things. But my intuition isn't helping me here.
1
vote
1answer
37 views

Natural logarithm equation, beginner stage

I am learning about natural logarithms and this is the first equation i must solve: $$ 30 - 23 e^{-0.027x} > 20 $$ Could somebody explain what i should do to solve this and other equations like ...
2
votes
3answers
103 views

What is the correct integral of $\frac{1}{x}$?

I understand that the graphs of $\log(x)$ and $\ln(x)$ both have derivatives (changes in slope) that follow the pattern of: $$\frac{d}{dx}\log_{b}x= \frac{1}{(x\ln(b))}$$ However, depending on the ...
6
votes
2answers
2k views

Intuition behind logarithm inequality: $1 - \frac1x \leq \log x \leq x-1$

One of fundamental inequalities on logarithm is: $$ 1 - \frac1x \leq \log x \leq x-1 \quad\text{for all $x > 0$},$$ which you may prefer write in the form of $$ \frac{x}{1+x} \leq \log{(1+x)} \leq ...
3
votes
3answers
62 views

Show that if $1> x>0$, then $x-1 ≥ \ln(x) ≥ 1−1/x$

Show that if $1> x>0$, then $x-1 ≥ \ln(x) ≥ 1−1/x$. I know the is using the MVT I can proof it for $x> 1$ but I don't understand how to proof for $x > 0$ .
2
votes
1answer
63 views

Integrating the logarithm of a function including a square root of a second degree polynomial

I have been trying for some time to calculate the following integral: $$\int \ln\left(k+\sqrt{ax^2+bx+c}\right)\ dx$$ where k, a, b and c are real numbers. I have tried several strategies, but without ...
1
vote
3answers
34 views

Differentiating this problem $\frac{2t^{3/2}}{\ln(2t^{3/2}+1)}$

How does one differentiate the function $$y(t)=\frac{2t^{3/2}}{\ln(2t^{3/2}+1)}.$$ I am still tying to understand MathJaX and not sure what is wrong with the expression. Anyways, How do I ...
1
vote
3answers
35 views

$N =\sum_{k = 1}^{1000}k(\lceil\log_{\sqrt{2}}k\rceil-\lfloor\log_{\sqrt{2}}k\rfloor). $

Find $N$ for $$N =\sum_{k = 1}^{1000}k\left(\left\lceil\log_{\sqrt{2}}k\right\rceil-\left\lfloor\log_{\sqrt{2}}k\right\rfloor\right)\;.$$ How could you solve this problem? Are there sigma rules or ...
0
votes
1answer
22 views

Strange log scale on a plot. How do I read this?

Doing an assignment with a strange log-log data plot. You'll notice that there at 14 segments per cycle, and they are not spaced as usual. Note the last 4 segments break the pattern of reduced ...
8
votes
4answers
3k views

Prove that if $a^x=b^y=(ab)^{xy}$, then $x+y=1$ using logarithms

Prove that if $a^x=b^y=(ab)^{xy}$, then $x+y=1$. How do I use logarithms to approach this problem?
1
vote
3answers
18 views

Let $ f(x)= ( \log_e x) ^2 $ and (Integration by parts. Comparing integrals of different limits )

Let $ f(x)=( \log_e x) ^2 $ and let $ I_1= \int_{2}^{12} f(x) dx $ , $ I_2= \int_{5}^{15} f(x) dx $ and $ \int_{8}^{18} f(x) dx$ Then which of the following is true? (A)$I_3 <I_1 < I_2 $ ...
18
votes
12answers
2k views

Alternative notation for exponents, logs and roots?

If we have $$ x^y = z $$ then we know that $$ \sqrt[y]{z} = x $$ and $$ \log_x{z} = y .$$ As a visually-oriented person I have often been dismayed that the symbols for these three operators ...
3
votes
2answers
127 views

Solve $-B \ln y -A y \ln y + A y- A =0$ for $y$

I would like to know if there is a (preferably closed-form) solution for $-B \ln y -A y \ln y + A y- A =0$ for $y$ Where $A, B \in \mathbb{R}^{+}$. I have reasons to think there isn't a closed form ...
6
votes
2answers
273 views

Contour Integral $ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$

I need help evaluating this with contour integration $$ \int_{0}^{1}{\ln\left(\,x\,\right)\over \,\sqrt{\vphantom{\large A}\,1 - x^{2}\,}}\,{\rm d}x $$ I am not sure as to how to work with the branch ...
1
vote
1answer
40 views

Understand Logarithm of Bar values manipulation step.

Currently I am learning Logarithm , but I can't understand the manipulation of the following Highlighted step how it comes How the result come after after ...
0
votes
0answers
17 views

Order of operations for log transformation

I am working with a large dataset of positive values with a positive skew. I will be using a Ln transformation in SPSS to normalize my dataset. However, I am not sure of the order of operations. For ...