Questions related to real and complex logarithms.

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-2
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0answers
236 views

Fermat's last theorem generalization [closed]

Conjecture: Let $g$ is a positive algebraic number greater than two, then the equation ($x^g+y^g=z^g$) doesn't have any solution, where ($x, y$ and $z$) are three distinct positive coprime integers ...
-4
votes
0answers
13 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
19 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 ...
1
vote
0answers
37 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 ...
2
votes
2answers
70 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 ...
4
votes
4answers
109 views

What does $d\log\left(\frac{y}{x}\right)$ mean mathematically?

I am used to seeing derivatives written as $$\frac{df}{dx}.$$ But my economics professor keeps using notation like $$ d\log\left(\frac{y}{x}\right)$$ and I have no idea what this means. What does ...
0
votes
2answers
24 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
14 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 ...
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!! $$x\log (54) +3\log (54) = x$$
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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
45 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 ...
1
vote
1answer
24 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 - ...
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
14 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
85 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.
0
votes
3answers
38 views

Solve this equation: $\log_3(3-2\cdot3^{x+1})=2+2x$

Solve this equation: $\log_3(3-2\cdot3^{x+1})=2+2x$. I put $(2+2x)^3=3-2\cdot3^{x+1}$. But I don't know how to go on.
3
votes
2answers
134 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
58 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
2
votes
1answer
76 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
0answers
51 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 :)
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
3
votes
3answers
120 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
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 ...
3
votes
4answers
188 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: $$ 2^{n-3}=\frac {20}{n} $$ ...
-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( ...
-6
votes
0answers
48 views

Logarithms $\log_8\boxed!+\log_87=\log_877$ [closed]

$$\begin{align} \log_211-\log_25&=\log_2\boxed!\\ \log_8\boxed!+\log_87&=\log_877\\ 5\log_62&=\log_6\boxed! \end{align}$$ I don't know where to start with this problem. The three ...
31
votes
3answers
659 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 ...
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 ...
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] ...
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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
2answers
532 views

Quick logarithm calculation

In coming up with an algorithm for finding log (10) base 2, these are my thoughts. I wanted to know if this makes sense and how could I truly make it more efficient. The requirements are strictly not ...
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
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
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
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?
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.
6
votes
2answers
3k views

What's the formula to solve summation of logarithms?

this is my first question here. I'm studying summation and everything I know is that: $\sum_{i=1}^n\ k$ is $\frac{n(n+1)}{2}\ $ $\sum_{i=1}^{n}\ k^2$ is $\frac{n(n+1)(2n+1)}{6}\ $ $\sum_{i=1}^{n}\ ...
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 ...
1
vote
2answers
3k views

Compound interest - how to solve this with logarithms & geometric series?

I could use some help with the following: Jacques is saving for a new car which will cost 29000 dollars. He saves by putting 400 dollars a month into a savings account which gives 0.1% interest ...
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 $\log d^2 = 2 \log d$ ...
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votes
4answers
1k views

Prove that $\log_2 3$ is irrational [closed]

Prove that $\log_2 3$ is irrational Seemingly simple homework assignment. Was never the best with logarithms, how would I go about proving?