# Tagged Questions

Questions related to real and complex logarithms.

74 views

### Find all values of $x$

Determine all real values of $x$ such that: $$\log_{2}(2^{x-1} + 3^{x+1}) = 2x - \log_{2}(3^x)$$ Let $u = 2^x$ and let $y = 3^x$ For ease, let $\log_{2}$ be represented by just $\log$ so: ...
510 views

### Summing reciprocal logs of different bases

I recently took a math test that had the following problem: $$\frac{1}{\log_{2}50!} + \frac{1}{\log_{3}50!} + \frac{1}{\log_{4}50!} + \dots + \frac{1}{\log_{50}50!}$$ The sum is equal to 1. I ...
38 views

### Simplifying a logarithmic expression.

I have: $\log xy + \log 2x^2 - 0.5\log 4y^2$ The unlike terms make it hard to see what can be done? Thanks.
135 views

### How do I construct a function $\operatorname{sog}$ such that $\operatorname{sog}\circ\operatorname{sog} = \log$?

Imagine a real-valued semilog function $\DeclareMathOperator{\sog}{sog}\sog$ with the property that $$\sog(\sog(x)) = \log(x)$$ for all real $x>0$. My questions: Does such a function ...
15 views

### Why does the sawtooth graph that uses cos(x) instead of sin(x) have a minimum value of -ln(2) when x is a multiple of pi?

So you know how the sawtooth function is $\sum _{n=1}^{\infty}\frac{\sin \left(n\left(x\right)\right)}{n}$, and that the minimum value approaches -2, right? So when I use cos(x) instead of sin(x) ...
44 views

### solve $y = \frac{A }{\frac{B}{\ln(y/y_0)} - 1} \frac{1}{x^2}$

I'm trying to express y as a function of x, using the following equation : $$y = \frac{A }{\frac{B}{\ln(y/y_0)} - 1} \frac{1}{x^2}$$ Can anyone help me ? Thanks !  - I originally attempted ...
35 views

### If $a=b\log b$, how does $b$ grow asymptotically?

If $a=b\log b$, how does $b$ grow asymptotically in terms of $a$? I think the answer should be $b=\Theta\left(\frac{a}{\log a}\right)$. I tried taking logs to get $\log a=\log b+\log\log b$, but it's ...
46 views

34 views

### How can I simplify $N^{\frac{e}{\sqrt{\log(N)}}}$

I am working through an algorithms workbook and I have the following equation: $$N^{\frac{e}{\sqrt{\log(N)}}}$$ I know I can simplify it somehow using the properties of logs and exponents but am a ...
37 views

### Prove that $-(p_1+p_2)\log{p_1+p_2} \leq -p_1 \log{p_1} - p_2 \log{p_2}$ provided that $p_1,p_2 > 0$

WTS: $$-(p_1+p_2)\log{(p_1+p_2)} \leq -p_1 \log{p_1} - p_2 \log{p_2} \> \> \forall \> \> p_1,p_2 > 0$$ Any hints on this? I've tried to set it up as a proof by contradiction, and ...
33 views

### Solving the logarithmic rational equation

I'm wondering there exist the way to solve the equation form of: $$\log f(x) + g(x) = c$$ where $f(x)$ and $g(x)$ are rational functions, $c$ is a constant. Is there any general(in closed form) ...
39 views

### Solving exponential equations like $2^{2x} - 3 \cdot 2^x - 10 = 0$

I have two equations that I'm not able to solve. I know the answers, but I can't get to them. $$(a) \qquad 2^{2x} - 3 \cdot 2^x - 10 = 0$$ (Answer: $x = \frac{\log 5}{\log 2}$.) On a) I ...
38 views

### Which functions have growth rates between $\log n$ and $n$?

Is there any function with a rate of growth between $n$ and $\log(n)$? My problem is that I have a value $x$, against which a term varies. The term does not vary as rapidly a linear function of $x$ ...
21 views

### Prove that $b^{\log_d(a)}=a^{\log_d(b)}$ and $\log_a(b)*\log_b(c)=\log_a(c)$

How to prove $b^{\log_d(a)}=a^{\log_d(b)}$ and $\log_a(b)*\log_b(c)=\log_a(c)$ I've found this properties but the book doesn't include the proofs of them. Can you help me please?
25 views

### How can I figure out what the log function being used based off the X and Y values?

I have a chart where Microsoft .NET has automatically scaled it using a (supposedly) log10 function of some kind. I need to figure out what formula they're using for the value at each tick mark. The ...
11 views

### When is a function a branch of another multi-valued function?

Definition (Branch): A branch of a multiple-valued function $f$ is $\color{teal}{\text{any}}$ single-valued function $F$ that is $\underline {analytic}$ in some domain at each point $z$ ...
45 views

### How can solve that log

How can solve that logarithms $\log _{\frac{4}{x}}\left(x^2-6\right)=2$ It's look diffucult to solve I was solve but stop with $x^4−6x^2−16=0$ what is next?
230 views

### Prove without using a calculator $(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}}<\pi$

Prove without using a calculator $$(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}}<\pi$$ I want to know if there is an easy way to prove this inequality without using a calculator.
42 views

### Find value of $x$ which satisfies the equation $\log_9 x=(\log_3 x)^2 ,x>1$

So far I got this $(\log_3 x)/2=(\log_3 x)(\log_3 x)$. Then I am stuck. Any idea or nice elaboration on this problem will be kindly appreciated.
23 views

### Can you take the natural log of a change variable?

My mathematics is quite rusty right now. Does it make sense to take the natural log of a change variable? I am taking the log of the change in employment between firms in 2005 and 2004. So I will ...
18 views

### Why does the branch cut for the principal branch of log(z+1) start at z=-1?

If I cut away the negative real axis to make log(z+1) single-valued, why does the branch cut start at z=-1 and not at the origin z=0? Why does the shift in argument from log(z) to log(z+1) make it ...
32 views

49 views

### How do I evaluate this without using taylor expansion :$\lim_{x \to \infty}x^2\log(\frac {x+1}{x})-x\$?

How do I evaluate this without using Taylor expansion? $$\lim_{x \to \infty}x^2\log\left(\frac {x+1}{x}\right)-x$$ Note: I used Taylor expansion at $z=0$ and I have got $\frac{-1}{2}$ Thank ...
42 views

### simplify a log expression

I met a problem, I don't know if this term can be simplified properly? $$e^{ (\ln ax^{b})^{c}}$$ since the ln term with power of c is hard to cope with, thanks for any help!
22 views

### Why do we use logarithms and how does it work?

I solved: We know the content of the evaporator (content in ml), the percentage of foam or gas lost every day (evap_per_day) and the threshold (threshold) in percentage beyond which the evaporator ...
56 views

### Closed form for ${\large\int}_0^1x\,\operatorname{li}\!\left(\frac1x\right)\ln^{1/4}\!\left(\frac1x\right)dx$

Let $\operatorname{li}(x)$ denote the logarithmic integral: $$\operatorname{li}(x)=\int_0^x\frac{dt}{\ln t}.$$ How can we prove the following conjectured closed form? ...
67 views

### How do you find the derivative of the integral $\sin(\ln x)$

$$\frac{d}{dx} \;\left[ \int_{a}^{x^2}\sin(\ln(z))\;dz\right]$$ I'm not sure if I'd have to do the chain rule on the natural logarithm and them $x^2$, or if there is no chain rule at all. Any help ...
39 views

### How to Find the Domain of The Inverse of (4(e^x)-5)/(25(e^x)+12)

These are the steps I have taken so far: In order to find the inverse of the function, I did the following steps: ...
82 views

### How can I simplify this expression of tricky logarithms?

How can I simplify this expression: $$\large \frac {2^n}{n^{nlog(2)}(log(2))^{nlog(2)}}$$ My guess is that somehow I can extract a $2^n$ factor from the denominator, but how can I do that? Would it ...
191 views

### Mistake on a Major Maths Website

I think I have found two massive errors on Math.com but I throw my logic out here for a third party to verify as I've been doing Contour Integrals for about 10 hours straight now so I am very tired. ...
121 views

### How to solve an equation with a tangent divided by a logarithm?

Here is an equation and I've never met this kind before. I would greatly appreciate your help. Maybe it's ridiculously simple and I overlook something? $$-12=\frac{\tan(x+4)}{\log(x+0.25)}$$
57 views

### Logarithmic inequality and properties of logarithms

So the inequality: $$2 \cdot \log_{\sqrt3}{(1-x)} - \log_\sqrt3 {(3-x)} \lt 2$$ Can be written as: $$\log_{\sqrt3}{(1-x)^2} - \log_\sqrt3 {(3-x)} \lt 2$$ ????????????? I have tried both on ...
53 views

### How to prove “Logarithms grow more slowly than polynomials”

"Logarithms grow more slowly than polynomials. That is, Θ(lgn) grows more slowly than Θ(n^a) for any positive constant a." if y1= x^(1/2) and ...
15 views

### optimal hill “rank” cannot be solved?

Okay so I was thinking about the following problem today: We have a guy who is h tall stand upon a paraboloid shaped hill of the form $z=-ar^n$ How far away (in r) does his friend who is also h ...
43 views

### Proving Logarithims

I'm trying to figure out how to go about proving this statement: $$n^{\log(a)} = a^{\log(n)}$$ I'm told that I cannot prove from both sides. I tried to $\log$ the first side to get: ...
20 views

### Big O proof for logarithmic function

I am an undergraduate student in Computer Engineering and going through one of the textbook examples, I am asked to prove that $T(n)$ is $O(\log{}n)$ Where $T(n)= 5\log_{2} (2n) +7$. I understand ...
47 views

### Proving $\rm \frac i 2 \ln\frac {x+i}{x-i}=\arctan x$ .

Proving $$\rm \frac i 2 \ln\frac {x+i}{x-i} =\arctan x$$ I'd like to prove this identity without taking the derivatives and integrating, what are some cool ways to prove this?
14 views

### Logarithm and exponential inverse (base 2)

if we have a function as follows $$\log_2(1+ S) \leq T$$ Which of the following is correct $$S\leq e^T-1$$ or $$S\leq 2^T-1$$ When should each be used, i.e. with what logarithm base? Thanks