Questions related to real and complex logarithms.

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2
votes
3answers
49 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 ...
0
votes
0answers
14 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 ...
0
votes
2answers
59 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
42 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 ...
-4
votes
0answers
14 views

Expressing index power.Hard [on hold]

2^r=3^s=6^t express t in the terms of r and s.
0
votes
0answers
22 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 ...
0
votes
2answers
25 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?
0
votes
0answers
15 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
vote
0answers
38 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
45 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
votes
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 ...
-3
votes
3answers
25 views

How to solve the following equation (xlog)? [on hold]

I have to review questions from my math test and I'm stucked at this one. Can somebody explain me how to solve it ? Thank you!! $$x\log (54) +3\log (54) = x$$
-12
votes
0answers
47 views

How to solve xlog54 +3log54 = X? [on hold]

I have to review questions from my math test and I'm stucked at this one. Can somebody explain me how to solve it ? Thank you!!
2
votes
1answer
46 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
76 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 ...
0
votes
0answers
15 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
86 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
vote
1answer
26 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
136 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 + ...
-4
votes
0answers
34 views

how to simplify $\sqrt{\cos (x)} \sinh \left(\ln (2) x^{\cos(x)}\right)+\sqrt{\cos (x)} \cosh \left(\ln (2) x^{\cos(x)}\right)$ [duplicate]

$\sqrt{\cos (x)} \sinh \left(\ln (2) x^{\cos(x)}\right)+\sqrt{\cos (x)} \cosh \left(\ln (2) x^{\cos(x)}\right)$ = $2^{x^{\cos(x)}}\sqrt{\cos(x)}$ if $x > 0$ and $\cos(x) > 0$? Can $\pi$ be ...
0
votes
1answer
59 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
2answers
86 views

Simplify $(\cos x)^{2^{x^{\cos x}}}$ with respect to $x$ & $pi$ [closed]

Simplify $(\cos x)^{2^{x^{\cos x}}}$ with respect to $x$ & $pi$... if $x > 0$ and $cos(x)$ $> 0$
2
votes
1answer
77 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
votes
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
43 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
votes
2answers
28 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 $abc=\mu$ ...
-1
votes
1answer
52 views

How do I Simplify this Logarithmic Expression? [closed]

Here is the expression: $$4\log(x) - \frac{2\log(x-2)}{3\log(x)}$$ How would I simplify it into one logarithm? Thanks in advance :)
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
61 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
-2
votes
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
36 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
189 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
26 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
52 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
0answers
22 views

Logarithm Question, please help. [closed]

Write each expression as a sum or difference of multiples of logarithms. Assume that variables represent positive numbers. $\log_3x^{2}(x-9)$
0
votes
0answers
27 views

I can't find how to do solve this Logarithmic Problem [closed]

Hey I Can't get any idea about How to solve Below's Logarithm problem. Can someone please help me ? Here is the question : If $$x^2 + y^2 = 7xy$$ then prove that $$\log\left[\frac{1}{3}(x+y)\right] ...
3
votes
3answers
121 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 ...
0
votes
2answers
28 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
votes
1answer
27 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
56 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?
0
votes
1answer
13 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?
1
vote
1answer
40 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
votes
1answer
29 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
98 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
12 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 ...
1
vote
2answers
39 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
23 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, ...
3
votes
3answers
43 views

Logarithmic Differentiation equation, Help!

So, I have to differentiate this via $\log$. I am still learning, so please be patient, I will try to explain everything I did. Please tell me if it is correct. ...
31
votes
3answers
666 views

Integrals of the form ${\large\int}_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx$

I'm interested in integrals of the form $$I(a,b)=\int_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx,\color{#808080}{\text{ for ...