Questions related to real and complex logarithms.

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0answers
26 views

Multiple polylogarithms

$1)$ If $$I(a_o;a_1 \dots a_n ; a_{n+1}) = \int_{a_o}^{a_{n+1}} \frac{dt}{t-a_n}I(a_o;a_1\dots a_{n-1};t)$$ and $G(a_1 \dots a_n;z) = I(0 ; a_n \dots a_1;z)$, where $G$ is a multiple polylogarithm, ...
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3answers
205 views

How to find the inverse of $n\log n$?

So I'm on chapter $1$ of introduction to algorithms & at the end the book proposes a problem: here The answers are there & I was able to work through most of them myself despite my lack of ...
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1answer
16 views

The domain of logarithmic functions with sinx and cosx

I need to solve this equation: $\log_{\cos x} \sin x + \log_{\sin x} \cos x=2$ But in order to solve it, I first need to find the domain. What I did was this: $\cos x\neq1 \wedge \sin x\neq1 $ ...
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1answer
36 views

Polylogarithms and the shuffle algebra

$1)$ Write $\text{Li}_2(1-\frac{1}{x})$ in terms of $\text{Li}_2(x)$ and logarithms by considering its integral representation and suitable changes of variables. Attempt: The di-log is defined as ...
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2answers
44 views

Logarithm with nth root [closed]

I made it but the result is very strange. I want every step to the result $$ \large 6\log_{10}\frac{\sqrt2}{\sqrt[3]{3+\sqrt5}} $$
7
votes
4answers
493 views

Doubt about the domain in logarithmic functions.

According to my book, the logarithmic function $$\log_{a}x=y$$ is defined if both $x$ and $a$ are positive and $x\neq 0$ and $a\neq 1$. So are these not correct? $$\log_{-3}9=2$$ $$\log_{-2}-8=3$$ ...
1
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1answer
27 views

tight estimate for a log-linear inequality

Given $q>0$ and $p$, how do we get a tight estimate for the smallest $x$ such that $x\log(x)+px \geq q$? (such an $x$ always exists).
2
votes
1answer
98 views

Are logarithms radicals? [closed]

Does the set of all logarithms with a radical base and argument belong to the set of all radicals? A simple yes, no answer will suffice, an explanation would be wonderful. EDIT 1 Can a logarithm with ...
1
vote
2answers
83 views

derivative of $\ln(4)$

what is the derivative of $\ln(4)$? I am trying to find the derivative of this equation: $h(x)=\ln(\frac{x^3\cdot e^x}{4})$ by rules of logs I simplified the $h(x)$ to the following: ...
2
votes
3answers
53 views

Values of a for which equation $\log_ax = \lvert x+1 \rvert + \lvert x-5 \rvert$ has a unique solution

\begin{equation*} \log_ax = \lvert x+1 \rvert + \lvert x-5 \rvert. \end{equation*} I don't even know how to approach this one, any hints would be amazing. I tried separating into two cases, where ...
1
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1answer
29 views

I would like to know how to do log transformation of hyperparameters in Gaussian Process Classification.

I am using Gaussian Process classification and I want to do log transform of the hyperparameters so that they are all positive. From this www.lce.hut.fi/research/mm/gpstuff/GPstuffDoc.pdf document, I ...
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2answers
60 views

Solutions of $2^x 7^{1/x}\le 14$

The solution is supposed to be $(-\infty,0)$ and $[1,\log_2 7]$. What I get when solving the problem is $(-\infty, \log_2 7]$. Where did I get it wrong? I start by dividing both sides by 14, then ...
0
votes
1answer
50 views

logarithmic Series

I'm aware that by properties of logarithm $$\sum_{k=1}^n \ln (k) = \ln (n!)$$ My question is if $$\sum_{k=1}^n \ln^2 (k) = \ln^2 (n!)?$$ Because when I am verifying the value where $n = 5$, I get ...
0
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0answers
27 views

Discrepancy on the standard deviation of logarithmic function

Good day, Sir/Madame! I'm currently working on the standard deviation of a particular function $\frac{2}{\pi} \ln n$, where n is the degree of certain random polynomial. By the use of computer ...
2
votes
1answer
31 views

How to prove that the dependent variable could not be expressed explicitly in terms of the independent variable(s)?

Consider the equation that $$xy=\log{y}+1\text{.}$$ How does one prove that $y$ cannot be expressed explicitly in terms of $x$? By the way, I do not know how the adverb "explicitly" is strictly ...
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2answers
27 views

SUmmation of natural logarithm [duplicate]

Good day! Is there a formula that approximate the summation of natural logarithm of N as N runs from 1 to infinity?
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0answers
18 views

Estimation for a logarithmic function in $(0,\,1)$. A series should be used?

Let $f(t)\geq C_1t^{-\alpha}$ for all $t\in(0,\,\infty)$ and for some $C_1>0,\,\alpha>0$. and let $g(t)\geq C_2\left(\ln(t^{-1})\right)^\beta$ for all $t\in(0,\,1)$ and for some ...
1
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0answers
43 views

Solve $x=C \log(C \log(x+A)+B)$

Is it possible to resolve an equation of the type $$x=C\log{(C\log{(x+A)}+B)}$$ (where $A$, $B$, and $C$ are real-valued parameters) for $x$? As far as I can see, the function on the right hand ...
0
votes
1answer
50 views

Logarithm problem

If $a^x=b^y$, then how come $x\log a=y\log b$ holds? Can anyone show me how this is with all steps and necessary logarithm formula?
2
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2answers
63 views

what will be the value of this integral

$$ \large{ \int^{\Large{\frac{\pi}{2}}}_{0} \left[ e^{\ln\left(\cos x \cdot \frac{d(\cos x)}{dx}\right)} \right]dx}$$ We know that $\large{a^{log_a(c)} = c}$. But in this question, the expression in ...
2
votes
1answer
49 views

Is this manipulation with logs allowed?

$$\left( \frac{6}{7} \right) ^n < \frac{1}{65}$$ The answer is, by looking at which way the sign should be round: $$n > \log_\frac{6}{7}{\left(\frac{1}{65}\right)} \implies n>\frac ...
2
votes
3answers
85 views

L'Hôpital's rule exercise with natural log function

I'm looking for some advice on the following exercise: $$\lim_{x \to 0^+}{\ln{(\frac{1}{x}})}^x$$ This is my work so far: $$\lim_{x \to 0^+}{\ln{(\frac{1}{x}})}^x = \lim_{x \to ...
1
vote
1answer
16 views

Compound Interest Calculation

In __________ years a sum will double at $5\%$ per annum compound interest. Options given are: a. 15 years 3 months b. 14 years 2 months c. 14 years 3 months d. 15 years 2 months The way to ...
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0answers
18 views

Troubles understanding task for complex logarithm.

I have troubles understanding this question and what to do, the goal is to show that there is no complex determination of the logarithm and square root and those two are just some parts of the whole ...
6
votes
3answers
90 views

Solve $6^{x+8} = 4^{x-1}$

I tried doing $log_6\left(6^{x+8}\right) = log_6{4^{x-1}}$ I got stuck, and I don't think that was the right route.
1
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1answer
30 views

Upperbound a logarithmic expression that has a covariance matrix

Let $\Sigma$ be a $2\times 2$ covariance matrix and ${\bf h}$ a vector of complex values entries. $$A= \log(1+ {\bf h}^* \Sigma {\bf h} )$$ $$\Sigma = \begin{bmatrix} 1-|\rho_1|^2 & \rho_3 - ...
3
votes
2answers
147 views

Basic Logarithm question - I can't get both answers from quadratic

Here's the Question : If $xy$ = $64$ and $\log_x y + \log_y x = \frac{5}{2}$, find $x$ and $y$ I can get this to $$log_x y + \frac{1}{\log_x y} \frac{5}{2}$$ let $\log_x y = N$ $$N + ...
0
votes
1answer
70 views

$2^{x^{\cos(x)}}\sqrt{\cos(x)}$ can you rearrange mathematically to ${\cos(x)}\sqrt2^{x^{\cos(x)}}$ [duplicate]

$2^{x^{\cos(x)}}\sqrt{\cos(x)}$ can you rearrange mathematically to ${\cos(x)}\sqrt2^{x^{\cos(x)}}$ if $x > 0$ and $\cos(x) > 0$
2
votes
1answer
88 views

Integral with Logarithms

$$\displaystyle \int _{ 0 }^{ \pi /2 }{ \log(\cos(x))\log(\sin(x)) \ dx } = \dfrac { \pi { \ln}^{ A }(B) }{ C } -\dfrac { { \pi }^{ D } }{ E } $$ $$$$ This was one solution, but it went completely ...
0
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1answer
46 views

Proof of $\log^x{x} > x^{\sqrt{x}}$ for big $n$

How can I prove, that $$\log^x{x} > x^{\sqrt{x}}$$ for big $n$ ? I tried to logarithm those expressions, deduct them, somehow estimate the values but no luck. After few tries, I ended up with ...
2
votes
4answers
44 views

Gradient of a curve $y=\ln \sqrt{x+y}$

Find the gradient of the curve $y=\ln \sqrt{x+y}$ at the point when its y-coordinate is 1. My attempt, I differentiated and I got $\frac{dy}{dx}=\frac{1}{2x+2y-1}$. But I've problem in finding the ...
0
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2answers
30 views

Proving logarithm question

Prove: $$\log_a (bc)\times \log_b (ac)\times \log_c (ba)=2+\log_a (bc)+ \log_b (ac)+ \log_c (ba)$$ I took LHS and applied base change formula. I changed base to $`\text{abc'}$ Let $abc=\mu$ ...
0
votes
1answer
15 views

Interval of the solutions to $\log_{1/2}\log_2(\frac{1+2x}{1+x})>0$ is?

I consistently get $x>-1$ but that doesn't fit the possible solutions I've got. First step I do is state that $\log_2(\frac{1+2x}{1+x})<1$ Then express the $1$ as $\log_22$ and so on. What ...
2
votes
7answers
63 views

Another combined limit

I've tried to get rid of those logarithms, but still, no result has came. $$\lim_{x\to 0 x \gt 0} \frac{\ln(x+ \sqrt{x^2+1})}{\ln{(\cos{x})}}$$ Please help
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2answers
40 views

Integral with logarithm is positive

Given the following integral: $$I(f) = \int_\mathbb{R} f(x) \log \left(f(x) \sqrt{2\pi} e^{\frac{x^2}{2}}\right) dx,$$ where we assume $\int_{\mathbb{R}} f(x)\, dx =1$ and $f\geq 0$ a.e. Assume for ...
1
vote
1answer
40 views

How to solve for x in $2^{2x^2}+2^{x^2 + 2x + 2} =2^{5+4x}$

This is the question: $$\large{2^{2x^2}+2^{x^2 + 2x + 2} =2^{5+4x}}$$ What I did was put $~\large{2^{x^{2}}=t}$ From this, I got, roots of the quadratic: $$\large{-2^{x+1}\pm~\left( ...
3
votes
4answers
234 views

Solving equations with exponentials and a non-exponential term.

I know how to solve exponential equations. Just use logarithms, e.g., $$ 2^x-3=0 \\ 2^x=3 \\ x=log_23 \\ $$ But on a recent math test I found an equation of the form: $$ 2^{n-3}=\frac {20}{n} $$ ...
1
vote
0answers
28 views

Proof $\log(cn)$ is in $\Theta(\log(n))$

How can I prove that $\log(cn)$ is in $\Theta(\log(n))$, where $c$ is a constant? I tried to prove $c_1\log(n) \le \log(cn) \le c_2\log(n)$, where $c_1$ and $c_2$ are also constants, but I'm having ...
3
votes
4answers
54 views

Solve the equation $\log_{2} x \log_{3} x = \log_{4} x$

Question: Solve the equations a) $$\log_{2} x + \log_{3} x = \log_{4} x$$ b) $$\log_{2} x \log_{3} x = \log_{4} x$$ Attempted solution: The general idea I have been working on is to make them ...
3
votes
3answers
124 views

antiderivative of $\frac{1}{z(z-1)}$, complex logarithm

I have the domain $\mathbb{C} \backslash [0,1]$ and want to show that $$\int_\gamma \frac{1}{z(z-1)}dz = 0$$ for all closed curves $\gamma$. I want to accomplish this by explicitly finding an ...
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votes
2answers
29 views

need approach to solve given logarithm expression

I was going through algorithm on sorting and encountered a logarithm problem which need to be solved. Question Statement is: For inputs of size n, insertion sort runs in $8n^2$ steps, while merge ...
0
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1answer
36 views

Basic Logarithm equation, and how best to approach this question logically

Question: Solve the equation $$\log_3 \left(1 - 3x\right) = \log_9 \left(6x^{2} - 19x + 2 \right)$$ There's quite a bit going on, I'm trying to think about the best point to start in order to ...
0
votes
3answers
59 views

Changing the base of a logarithm

I must simplify $\log_4 (9) + \log_2 (3)$. I have tried but I can't get the correct answer $2 \log_2 (3)$. How do I proceed?
1
vote
1answer
18 views

on the convergence of an infinite series involving logarithms

It looks like the following quantity $$ q(k)=\frac{k+1}{2k}(1+\log k) - \sum_{i=2}^k \frac{i}{k^2} \log i $$ tends to $3/4$ as $k$ goes to infinity. Is there a nice way to prove it?
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1answer
41 views

Compute $(\ln(n!))^2$

In a discrete mathematics past paper, I must solve the following problem: We know (from the Stirling approximation) that ...
0
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1answer
32 views

Simplifying Logs

Simplify: $$\frac{\log a + \log b - \log c}{\log d^2}$$ Using the basic properties of logs, the numerator should simplify to $\log (ab/c)$, if I'm not mistaken. The denominator $\log d^2 = 2 \log d$ ...
4
votes
3answers
104 views

$ \frac{1}{2} + \dots + \frac{1}{n} \le \log n $

could anyone give me any hint how to prove this ? $$ \frac{1}{2} + \dots + \frac{1}{n} \le \log n $$ just came acroos this expression in my book.
1
vote
1answer
31 views

Finding a base given an exponent

In math, the logarithm of a number $n$ in base 10, finds the exponent where 10 has to be raised to, to produce $n$ again. So if $Log_{10}(n) = p$ then $10^p = n$. What I'm looking for is essentially ...
2
votes
2answers
45 views

How to prove this logarithm equation?

Given : $$\log_{12}18 = a \text{ and }\log_{24}54=b$$ prove that: $$ab + 5(a-b) = 1$$ My attempt: I couldn't solve it in any way, as base were not common. I could solve it if base of second ...
3
votes
2answers
27 views

Basic simultaneous equation with logarithms

Question Solve giving your answers as exact fractions, the simultaneous equations : $$8^y = 4^{2x + 3} \tag{1}$$ $$\log_2 y = \log_2x + 4 \tag{2}$$ I think that the RHS of eq 1 can be split up, ...