Questions related to real and complex logarithms.

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2answers
25 views

Expanding logarithm of function

Is there a way (there has to be), I can expand an expression like this? $$\log_2 (3f(n)^n)$$ P.S. This part of an assignment I'm working on, please do not give solutions
1
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2answers
32 views

Find all $x$ such that $8^x(3x+1)=4$

Find all $x$ such that $8^x(3x+1)=4$,and prove that you have found all values of $x$ that satisfy this equation. My effort Rewriting the equation I have \begin{array} 22^{3x}(3x+1)&=2^2 \\ ...
-4
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1answer
43 views

solve $4x^2y'' + y=0$, $y(-1)=2, y'(-1)=4$

This would require taking the $\ln(-1)$, which Zill solved in the 7th edition of diff eq $4.7$ problem $37$ by substituting $t$ for $x, y(1)=2, y'(1)=4$. Then substituting $-x$ for $t$ in the final ...
2
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2answers
23 views

How to express $\log_5 2$ in terms of a and b (Refer to qn)

In my textbook, I came across this interesting question which I am currently struggling to solve: If $\log_6 2 = a$ and $\log_5 3 = b$, express $\log_5 2$ in terms of a and b The solution given is ...
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1answer
22 views

Prove that $\sum_{i=0}^{k} \lg \frac{n}{2^i} = \Theta(\lg^2 n)$

Show that if $n$ is a power of $2$, say $n = 2^k$, then we have the equality $\sum_{i=0}^{k} \lg \frac{n}{2^i} = \Theta(\lg^2 n)$. The first step is to prove $O(\lg^2n)$: $$ \lg \frac{2^k}{2^0} + \lg ...
1
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3answers
61 views

Prove $\ln x \ge \frac{x-1}{x}$

Prove that for every $x>0$: $$\ln x \ge \frac{x-1}{x}$$ What I did: $$f(x) = \ln x, \text{ } g(x) = \frac{x-1}{x} $$ $$f(1) = g(1) = 0 $$ So it's enough to prove that $$ f'(x) \ge g'(x)$$ ...
4
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3answers
421 views

Find root of the equation

Find maximum root of the equation $$x - \frac{1000}{\log 2} \log x = 0$$ It locates between $13746$ and $13747$, but I want to find right solution not using graphing calculators. Thanks in advance.
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5answers
38 views

Equation $\log(x^2+2ax)=\log(4x-4a-13)$ has only one solution; then exhaustive set of values of $a$ is

Equation: $$\log(x^2+2ax)=\log(4x-4a-13)$$ It has only one solution; then exhaustive set of values of $a$ is ?? I don't even know where to begin The answer is : $$(-13/4,-13/12) \cup [-1]$$
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1answer
23 views

Solving for numerator in equation with logarithms (Activation Energy Equation)

I'm having trouble solving for k1 in this equation: ln(0.286/k1) = (100000/8.314)(1/500 - 1/490) The right side should equal 0.491, which I can calculate just fine, but then the left side gives me ...
0
votes
1answer
27 views

Converting equation to slope-intercept form

It's been awhile since I've worked problems like these and I am a bit hazy on some of the rules. I was hoping someone could show me how these are solved so that I can make sure I'm on the right path: ...
5
votes
1answer
146 views

Proof that $\frac{2}{3} < \log(2) < \frac{7}{10}$

Positive integrals $$\int_{0}^{1}\frac{2x(1-x)^2}{1+x^2}dx=\pi-3$$ and $$\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi $$ (http://math.stackexchange.com/a/1618454/134791) prove that ...
0
votes
2answers
24 views

Most logical thing to do with these exponents and sums?

I'm doing homework for a programming class and came across this problem. There's no directions besides what I've shown, so I don't even know what it's asking me to do. What makes the most sense for ...
1
vote
1answer
38 views

Closed-form Solution of Log Sum

I have the series: $$\sum_{i=1}^{i=10^N} \log_5 i$$ I'm trying to figure out how to get the closed-form solution to this problem. I entered it into WolframAlpha and got that it equals: $ ...
0
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0answers
8 views

Calculating Data Rate using Quadrature Amplitude Modulation (QAM)

I was working on my telecommunications homework and I have these questions: Calculate the data rate for a 2400 baud signal where each symbol can take on one of two levels (M=2) Calculate the data ...
0
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1answer
23 views

solve and skecth $\log{|z|}=-2\arg(z)$

Ive asked this question a week ago, but nobody managed to answer but it is doing my heading from then. I know usually You demand some initial work done on the question but I just dont know how to ...
4
votes
1answer
68 views

How to solve $\ln(y)=\ln(x)e^{\ln(x+1)} $ for x?

I know that if I have had $y = x^{x+0} $ aka $y = x^x$ I could do $y = x^x$ // $x = e^{\ln(x)}$ $y=x^{e^{\ln(x)}}$ // $\ln$() $\ln(y) = \ln(x)e^{\ln(x)}$ then using Lambert's W function I ...
0
votes
3answers
62 views

$\frac{\ln(x^2)}{\ln(x)} = 2$? Why?

$\frac{\ln(x^2)}{\ln(x)} = 2$? Upon trying to evaluate $\frac{\ln(x^2)}{\ln(x)}$, i've found that google plots it as always equal to 2, other than 0 where it is undefined. Why is this the case?
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2answers
27 views

Solving an exponential equation with x as a base and an exponent

So here's the problem: $x+3=3^x$ Obviously, graphing both sides and finding the intersection would reveal the answer, but algebraically, how can this be solved?
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0answers
21 views

Logarithmic function transformations

The standard log function form is $a \log[k(x-d)] + c$ Where $a$ vertically stretches or compresses $k$ horizontally stretches or compresses $d$ translates left or right $c$ translates up or ...
1
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1answer
51 views

How do you differentiate the integral from $ \int_{e^{-x}}^{e^x} \sqrt{1+t^2}\,dt$ [duplicate]

How do you differentiate the integral from $e^{-x}$ to $e^x$ of $\sqrt(1+t^2)$ with respect to t? $$ \int_{e^{-x}}^{e^x} \sqrt{1+t^2}\,dt $$ I know the answer is $$ e^x\sqrt{1+e^{2x}} + ...
0
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1answer
41 views

Does $\log_2 \sqrt[4]4$ exist?

Tomorrow I have an exam about graphics and log operations. Our teacher gave us a paper with exercises to practice and one of the exercises is: $\log_2 \sqrt[4]4$ I couldn't find the solution. ...
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2answers
31 views

Order of growth of logarithms, compared to linear

I think it is true that any power of a logarithm, no matter how big, will eventually grow slower than a linear function with positive slope. Is it true that for any exponent $m>0$ (no matter how ...
2
votes
2answers
34 views

Solve for $x\quad \log_2(2^n) = \log_2(1+x)$

I am out of practice with logs, but this is derived from the channel capacity theorem. $$B\log_2\left(1 + \frac SN\right)$$ Solve for $x $ $$\log_2(2^n) = \log_2(1+x)$$ I need this equation ...
0
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3answers
113 views

Exponential (to the power of a logarithm) [on hold]

How do I solve the following equation: $(3x)^{ln3}=(4x)^{ln4}$ Thanks in advance!
0
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1answer
38 views

Prove that $\sum{\log_{ab^2c^2}a = \frac35}$ only if $a=b=c$ [on hold]

So I have $a,b,c \in (1, \infty)$. Prove that $$ \log_{ab^2c^2}a + \log_{a^2bc^2}b + \log_{a^2b^2c}c = \frac35 \Leftrightarrow a=b=c $$
1
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1answer
31 views

Derivative of matrix logarithm with respect to matrix

I saw in this post that $\frac{d}{dt}\text{logm}(Z(t)) = \frac{dZ(t)}{dt}(Z(t))^{-1}$ Is this true to say: $\frac{d}{{dU}}{\mathop{\rm logm}\nolimits} (A) = {A^{ - 1}}\frac{d}{{dU}}A$ where U is ...
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2answers
25 views

Difficulty finding the sum of a hyperbolic function.

Can someone please point out where I am (If I am) going wrong during the solution process of the following question: I have been presented with the following : $$4sinh(2ln(2))-cosh(ln2)$$ and told ...
1
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2answers
43 views

taking the natural log of e^(2x) = (4/3)

I have been unable to answer the following question. I must solve for x: $$e^{2x} = (4/3)$$ I have been made aware that I must take the natural log of both sides, giving: $$ln(e^{2x}) = ln(4/3)$$ ...
1
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2answers
86 views

Use $\log(x)$ to calculate $\log(x+1)$

Given that I know the value of $\log(x)$, I would like to calculate the value of $\log(x+1)$ on a computer. I know that I could use the Taylor expansion of $\log(1+x)$, but that uses $x$ rather than ...
0
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1answer
14 views

Logspace() in Matlab [closed]

In Matlab , Logspace() Generate logarithmically spaced vector . But what do we mean by them ? Earlier , I used to think that they are just equal spaced and their 10th power is returned . Like here : ...
0
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3answers
54 views

Show that $\frac{1}{1+k}=\frac{\frac{1}{k}}{1+\frac{1}{k}}\leq \ln(1+\frac{1}{k})\leq\frac{1}{k}$

Prove the following: $$\frac{1}{1+k}=\frac{\frac{1}{k}}{1+\frac{1}{k}}\leq \ln(1+\frac{1}{k})\leq\frac{1}{k}$$ I know I can prove it with induction if the values were naturals. However, the "problem" ...
1
vote
1answer
13 views

Log transformations of function domain and inequalities

If I know that for some function $f$, the following is true for $x, y \geq 0$: $f(\log (x^a y^b)) \leq f(\log x)^a f(\log y)^b$ Can I make the claim that $f(x^a y^b) \leq f(x)^a f(y)^b$ If I ...
3
votes
7answers
72 views

Evaluating $\lim_{n\to\infty}{n\left(\ln(n+2)-\ln n\right)}$

I am trying to find$$\lim_{n\to\infty}{n\left(\ln(n+2)-\ln n\right)}$$ But I can't figure out any good way to solve this. Is there a special theorem or method to solve such limits?
0
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0answers
21 views

Applying Cauchy-Riemann to $f(z)$

$$\ln|z|+i\text{Arg}(z)$$ the problem states that I have to apply Cauchy Riemann to the problem and determine a conclusion. Below is how far I got, but I'm not sure how to take the derivative of ...
0
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0answers
24 views

How to solve inequalities where the $x$ term appears inside the argument of multiple different functions?

We're asked to study the sign of the following function: $$\frac{x(\ln{x}+1)^2 - 2(\ln{x}+1)^2 - \frac{4}{x(\ln{x}+1)}}{(x(\ln{x}-1)^2)^2} \geq0,$$ in which the $x$ variable appears both outside ...
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2answers
48 views

Calculate Ln$(i^i)$

Calculate Ln$(i^i)$ My attempt: Ln$(z)$=$\ln|z|+i\arg z$ $$z=0+i^i=0+i\cdot i$$ $$|z|=\sqrt{0^2+i^2}=i\\ \arg z=\arctan(i/0)$$ $1.$ how it can be that the modulus equal to $i$? $2.$ how ...
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1answer
29 views

How to solve for a variable in logarithms

How do I solve this for $y$? $$u= 1 - \exp\left\{-\left(\frac{y-\theta}{\alpha}\right)^\gamma\right\}.$$ If I take the $\log$ I end up with $$\log(1-u) = ...
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0answers
36 views

Integral with logarithmic residue (Residue general method)

I was reading about this post, about Integration in complex plane with logarithmic residue: Integral with logarithm - residue And I thought about: what if $R(x)$ function is inside the logarithm? ...
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0answers
47 views

Defining a continuous complex logarithm on open set $U \subset \mathbb{C}$

Suppose you are given an open set $U \subset \mathbb{C}$ and a continuous function $f: U \rightarrow \mathbb{C}-\{0\}$. And $f$ has the next property: For every closed loop $ c: I \rightarrow U$ ...
0
votes
2answers
51 views

Prove $\log(x)^{10} < x$ (for $x > 10^{10}$)

I need to prove that $\log(x)^{10} < x$ for $\ x>10^{10}$ It's pretty clearly true to me, but I need a good proof of it. I tried induction, and got stuck there.
1
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2answers
43 views

How to solve this equation using logs

How do solve this equation for x using logarithms? $$4^x = 6^x-3$$ If it is not possible using logarithms, please provide another way. Thank you in advance
3
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1answer
31 views

Summation of $\sum_{k=1}^{n}\left \lfloor \log _{m}k \right \rfloor$ and $\sum_{k=1}^{n}\left \lceil log_{m}k\right \rceil$

$$\sum_{k=1}^{n}\left \lfloor \log _{m}k \right \rfloor$$ $$\sum_{k=1}^{n}\left \lceil log_{m}k\right \rceil$$ I found myself stuck trying to solve these two summations but i can't make any progress. ...
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0answers
24 views

How to expand in terms of inverse logarithms?

I'm currently working with a matched asymptotic expansion problem. Currently, I have a function $f$ that can be expanded as: $$f = f_0 + \frac{f_1}{\ln{\epsilon}} + \frac{f_2}{(\ln{\epsilon})^2} + ...
2
votes
2answers
57 views

True of false: The sum of this infinite series. [duplicate]

I'm fairly sure it is false, but I'm not quite sure about which test I should use to prove it. $$\sum_{n=2}^\infty \ln\left(\frac{n-1}{n}\right) = -1 $$ I think using the integral test should work, ...
0
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1answer
27 views

Calculating a Hausdorff Dimension from Formula

I am having difficulty computing the following formula for the Hausdorff Dimension for a type of Moran set called a partial homogenous Cantor set. ...
2
votes
2answers
29 views

Common factor out from a sum of exponential functions

From the below equation, which is a sum of two exponential functions I would like to compute the common factor $n$ $$ d = \exp\left(\frac{-x}{n}\right)+\exp\left(\frac{-y}{n}\right)$$ Unfortunately, ...
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0answers
39 views

Integral of complex logarithm on a disk in the plane

Let $a$ be a complex number and $D$ the disk centered around $0$ and of radius $R$. I would like to compute the integral I=$\int_D \log(|z-a|)d^2z$. I am interested in particular in the case $R\gg ...
22
votes
5answers
1k views

Solving a logarithmic equation that has an exception to the power rule

Given the following: $$\log_3({x^2-3})^2=2$$ If I were to use the power rule, I would do: $$2\log_3({x^2-3})=2$$ $$\log_3({x^2-3})=1$$ $$3^1=x^2-3$$ $$3+3=x^2$$ $$x=\pm\sqrt6$$ Substituting ...
3
votes
0answers
45 views

Derivative of the Logarithm - Dirac

So I stumbled across P.Dirac's book Principles of Quantum Mechanics and I found something really peculiar on page 61 of the Fourth Edition. He states that usually we accept that ...
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1answer
24 views

Smart way to calculate floor(log(x))?

I thought of an algorithm that involves $\lfloor \log_{b} x \rfloor$ and am trying to determine its computational complexity. At first glance my algorithm looks polynomial, but I read that my ...