Questions related to real and complex logarithms.

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0
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0answers
15 views

Baby-step Giant-Step algorithm to calculate value in new base

Using the Baby step–giant step algorithm I am trying to determine $log_{2}(7)$ in base $1$3. Let $p = 7$. Set $n$ to the least integer greater than $\sqrt p$: $n = 3$. So for baby step, I started off ...
0
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1answer
40 views

For which values of $a$ does this equation have a solution(s)?

The equation in question is $$\log_5x*(\log_5(2*\log_{10}a-x)*\log_x5+1)=2$$ Tried working this down with the rules of logarithms, got it down to a quadratic equation of $x$ with $a$ as one of its ...
0
votes
2answers
36 views

Log of many Logs

How can I compute the values of $n$ for which the following expression exists? $$\log_e(\log_e(\log_e(\log_e(\ldots\log_e(n))))$$ It is for instance apparent that when $n = e$, the second ...
1
vote
1answer
49 views

Integral of ln (3x) / x

I believe this should be a simple problem but I don't have an answer key to confirm if this is right, and some of the similar questions I can find online seem to be giving more complicated solutions. ...
-1
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3answers
52 views

Value of $x$ when $5 + \log x = \log \left(x^6\right)$

Find the value of $x$ when $$5 + \log x = \log \left(x^6\right)$$ I've tried many times to solve this, however I can't seem to find a correct (consistent) answer. My solutions range from $$x = e, x ...
1
vote
1answer
38 views

Simple Logarithmic question.

I was just wondering if i can do this. Q. Solve $\log_{9}24=x $ $\implies9^x =24$ $\implies3^{2x}=2^3 3$ $\implies\log_3(3^{2x})= \log_3(2^3 3)$ $\implies2x=2 (3)^{1/3}$ $\implies x=3^{1/3} $ ...
1
vote
2answers
66 views

Taking an infinite number of logarithms

Let $n$ and $k$ be two integer parameters ($n\geq k$, if that matters). Define the following function: $\text{LOG }x=\max(\log{x},1)$ What is the limit of the following sequence as a function of $n$ ...
1
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3answers
45 views

How do I find the critical points of this function involving e?

I have the function: $$g(x)={{1 \over \sqrt{2 \pi}} \cdot e^{{-(x-2)^2}/2}}$$ Through very tedious differntion, I got to: $$g'(x) = {{{-(x+2)} \cdot {e^{{-(x-2)^2}/2}}} \over {2 \pi}}$$ Setting ...
0
votes
2answers
24 views

Converting log form of equation into linear form

I am trying to convert part of an equation from its log form into a linear form. Specifically, I am trying to convert $10^{4 log (x)}$, into $x^4$, but I'm really unsure of how to get from this first ...
1
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4answers
43 views

Prove that $\log_a(b)=\log(b)/\log(a)$

Prove that $$\log_a(b)=\log(b)/\log(a)$$ I don't know how to solve it, but I need to prove it so solve a problem.
0
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2answers
55 views

Prove that $\log_a(b)=-\log_b(a)$

Can you prove that: $$\log_a(b)=-\log_b(a)$$ I just thought that it should equal $$\frac{\log(b)}{\log(a)}.$$ but I don't think anything else.
1
vote
1answer
24 views

How do I simplify a Multivariable expression involving derivatives of logarithms?

I have this expression I got after a lot of calculation: $$\sigma =\frac{d\log\left(\frac{b(x,y,\rho)}{r(x,y,\rho)}\right)}{d\log\left(\frac{ 2 ...
0
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1answer
11 views

Logarithm Subject of Formula

$G_{dB}(f) = −10 \log_{10}(1 +\left(\frac f{f_3}\right)^2N)$. I will like to make $N$ the subject of the formula. Any lead on how to achieve this will be appreciated.
4
votes
1answer
85 views

If $\frac{x-1}{e^x-1} = y$ then $x=?$

I have following equation: $$\frac{x-1}{e^x-1} = y$$ I want to solve this equation such that I have the value of $x$ in the term of $y.$ i.e. inverse of the equation
1
vote
2answers
18 views

Limit log-sum of exponentials

I'm trying to compute the following limit: $$\lim_{\lambda \rightarrow \infty} \frac{1}{\lambda}\log\sum_{i=1}^n \exp[\lambda a_i]$$ I tried to solve with L'hoptials: $$= \lim_{\lambda \rightarrow ...
0
votes
1answer
29 views

Transpose exponential equation [on hold]

Could somebody please help with transposing the following equation to isolate x to the left side of the equation to solve for x? $$ y = 10^{1.830 \log(x)} + 2.686 $$
0
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1answer
35 views

Using log to take derivative of a function

Is it safe to say that if $\frac{d}{dx}ln(f)= g $ for some functions f and g, then $\frac{d}{dx}f = e^{g}$? Why or why not? (novice high schooler here)
1
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1answer
32 views

Combining log terms

I have this particular problem. We have to combine the log terms into a single log term: $$\dfrac{(2\ln a- \ln b - 5\ln c)}{2}$$ I did it in the following way : $$''~= \ln a -\frac{1}{2}\ln b - ...
1
vote
4answers
246 views

Natural log limit question

I have to find $$\lim_{n\to\infty}\left(\ln(n-1)-\ln(n)\right)$$ I'm pretty sure I need to solve this using the asymptotes. So if I use the rule for logs I can do lim (ln((n-1)/n)) and I know that ...
1
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1answer
33 views

Minimum value of function $f(x)=x+\log_2(2^{x+2}-5+2^{-x+2})$ out of 5 options

Minimum value of function $f(x)=x+\log_2(2^{x+2}-5+2^{-x+2})$ out of 5 options A : $\log_2(1/2)$ B : $\log_2(41/16)$ C : $39/16$ D : $\log_2(4.5)$ E : $\log_2(39/16)$ I just... don't know how to ...
1
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0answers
33 views

Is this chain of inequalities correct?

Is this chain of inequalities correct? If not how to make it works? $$\frac{\ln \left( 1+x^3+y^3 \right)}{\sqrt{x^2+y^2}} \le \frac{\left( x^3+y^3 \right)}{\sqrt{x^2+y^2}} \le \frac{ \left( ...
3
votes
3answers
65 views

How to integrate $x\ln(x+1)$?

I am trying to compute $\int x\ln (x+1)\, dx$. I tried integrating by parts and ended up with: $$\int x\ln(x+1)\,dx = \frac{1}{2}x^2\ln(x+1) - \frac{1}{2}\int\frac{x^2}{x+1}\,dx$$ but I'm stuck here.
1
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2answers
53 views

Is $\log_{\cos x}(1)$ defined at $x=0+2k\pi$? [duplicate]

I have an equation like this: $\cos(x) ^ {\sin(x)} = 1$ I thought I would solve it like this: $\cos(x) ^ {\sin(x)} = 1$ $\sin(x) = \log_{\cos(x)}(1)$ $\sin(x) = 0 $ $x = 0+k\pi$ But I'm ...
1
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1answer
23 views

Does the following series of transformations of inequalities holds?

I am to calculate limit of the function $f(x,y)$ i am trying to apply squeeze theorem. Is the following series of transformations of this inequality correct? If not how to do this correctly? i.e. are ...
1
vote
1answer
29 views

Logarithm multivariable limit $\frac{\ln(1+x^3+y^3)}{\sqrt{x^2+y^2}}$

Find multivariable limit $$\lim_{\left( x,y \right) \rightarrow (0,0)}\frac{\ln(1+x^3+y^3)}{\sqrt{x^2+y^2}}$$ I was trying to find and inequality i've found out that: ...
1
vote
1answer
34 views

Finding the time for an epidemic/computer virus to infect a population

Question: "Suppose a computer worm makes 2 copies of itself on another computer in one millisecond. Estimate the time that is needed to spread to a population of 1,000,000 computers" How would I ...
1
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0answers
23 views

1st order ODE separable

everyone! :-) I've a ODE question with I can't solve. It's here: ${dy\over dx} = {{xy + 2y-x-2}\over {xy-3y+x-3}} $ I tried the following: ${dy\over dx} = {{xy + 2y-x-2}\over ...
0
votes
3answers
33 views

How is this logarithmic identical transformation true? [closed]

$$x^{1-\log x}=1\Leftrightarrow(1-\log x)\log x=\log 1?$$ I don't know how it can be true.
0
votes
1answer
22 views

How is this identical transformation true $x^{1-\log(x)}=1\Longleftrightarrow \log x^{1-\log(x)}=\log1$?

How is this identical transformation true $$x^{1-\log x}=1\Longleftrightarrow\log x^{1-\log x}=\log1\text{ ?}$$ I thought to put both sides on log: $$\log x^{1-\log x}=\log1,$$ but then I don't know ...
0
votes
1answer
41 views

Evaluation of Spence's function.

Spence's function is defined as $${\rm Li}_2 (z)=- \int_0^z \frac{\ln(1-u)}{u} \, du $$ where $$z \in {\mathbb C} \setminus [1, \infty )$$ For $|z|<1 $ $${\rm Li}_2 (z)= \sum_1^ \infty \frac{ ...
0
votes
2answers
26 views

How to get the graph for $y= \log_{1/a} (x)$ from $y= \log_a (x)$?

I know that it is symmetric the Ox axle, but I can't prove it.
1
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3answers
40 views

Limit of $y \ln (x^2+y^2)$

I want to calculate limit of $\lim_{(x,y) \rightarrow(0,0)}y \ln (x^2+y^2)$. How to do that? From iterated limits i know that limit exists for certain, but how to show that it is equal to zero then?
0
votes
1answer
15 views

If the positive numbers x,y,z are in harmonic progression, then log(x+z) + log(x-2y+z) equals

If the positive numbers x,y,z are in harmonic progression, then log(x+z) + log(x-2y+z) equals a) 4log(x-z) b) 3log(x-z) c) 2log(x-z) d) log(x-z) How do i approach this problem? IF x,y,z are in HP, ...
1
vote
5answers
97 views

How to solve $\ln(x) = 2x$

I know this question might be an easy one. but it has been so long since I solved such questions and I didn't find a an explanation on the internet. I'd like if someone can remind me. I reached that ...
0
votes
1answer
21 views

Number of integral values of M

If $\log_3M=a_1+b_1$ and $\log_5 M=a_2+b_2$, where $a_1,a_2$ are natural numbers and $b_1,b_2 \in [0,1)$. If $a_1a_2=6$, then find the number of integral values of M. What so I do in the problem. I ...
3
votes
1answer
40 views

System of logarithmic equations

$$\log (2000xy)-\log x\log y=4$$$$\log(2yz)-\log y\log z=1$$$$\log(zx)-\log z\log x=0$$ The base is 10 everywhere. I tried opening the log with the sum formulae and then manipulating, but I got stuck. ...
0
votes
1answer
25 views

How to find the domain of the function $f(x) =\log_y a^2$?

To find the domain of the function $f(x) = \log_y a$ it's enough to check if the base (y) is greater than 0 and not equals to 1 and the number (a) is greater than 0. But what if we have a power of ...
-1
votes
0answers
11 views

GLM for normal distribution

$Y_i$~N$\left(\mu_{i,}\sigma^2\right)\space \mu^2=\alpha+\log \left(\beta_0+\beta_1x_i\right)\space \alpha\space is\space unkown$ how is this proved to be a Genralized Linear Model? My assumption ...
10
votes
4answers
170 views

Calculate the infinite sum $\sum_{1}^\infty \frac{\log{n}}{2n-1}$

I would like to calculate an asymptotic expansion for the following infinite sum: $$\displaystyle \sum_{1}^N \frac{\log{n}}{2n-1}$$ when $N$ tends to $\infty$. I found that the asymptotic expansion ...
2
votes
3answers
100 views

How could this be true $n=\log(e^n)$?

I am learning elementary logarithms. How could this be true $n=\log(e^n)$? I searched online and couldn't find any info on this, could anyone give me some clue?
0
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2answers
38 views

Even and odd functions | logarithm [closed]

show why this logarithm is an odd function? $$y = \log_2 (x-\sqrt{1-x^2})$$
0
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7answers
103 views

Given $\log 2$ and $\log 3$, compute $\log 120$ [closed]

Given that $\log2= 0.30103$ and $\log3=0.47712$, calculate (without using tables or calculators) the value of $\log 120$.
3
votes
1answer
56 views

Find integer $n$ that satisfies $(\lg n)^{2^{100}} <\sqrt{n}$ with $n > 2$

If $(\lg n)^{2^{100}} < {n^{1/2}}$, where $\lg$ is the binary logarithm, then $$(\lg n)^{2^{101}} < n$$ $$2^{101}\lg \lg n < \lg n$$ $$101 < \lg \lg n - \lg \lg \lg n$$ I don't know that ...
0
votes
2answers
25 views

Show $\\Log z_1z_2 \neq Log z_1 + Log z_2$. given $z_1 = i$ and $z_2 = -\sqrt 3 + i$.

Show by evaluating both sides that for $z_1 = i$ and $z_2 = -\sqrt 3 + i$, $\\Log z_1z_2 \neq Log z_1 + Log z_2$. Recall the definition: $\\Log z = Log |z| + iArg z$ Attempt: left side: $\\Log ...
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votes
3answers
63 views

Solve for X in a difficult exponential function [closed]

Solve for $X$ when $3^{x^x}=1000$ By hand please (without evaluating the intersection on the graph). How is it done?
2
votes
0answers
43 views
+50

Why are logrithms of trigonometric functions useful

I have noticed that in many trigonometric tables the logarithm of the trigonometric values are given. Why this is given and not the actual values of the trigonometric functions? For example, instead ...
1
vote
0answers
34 views

Generator of group, Computation of discrete logarithm

The prime number $p=67$ is given. Show that $g=2$ is a generator of the group $\mathbb{Z}_p^{\star}$. Compute the discrete logarithm of $y=3$ as for the base $g$ with Shanks-algorithm. Compute the ...
2
votes
5answers
84 views

How to properly solve this inequality $2^x < \frac{3}{4}$?

How to properly solve this inequality? $$2^x < \frac{3}{4}$$ I know that it will be something like that: $$ x \stackrel{?}{\ldots} \log_2\frac{3}{4} $$ But I don't know how to decide if it should ...
1
vote
2answers
35 views

What are these numbers in a logarithmic table?

Below is an image from a table of logarithms. As an example, one sees that $\log(661.3) = 2.82\color{red}{040}$. In this logarithmic table there are some numbers to the right. My question is: What is ...
0
votes
2answers
43 views

Prove that if $x>1$, then $log_a(x)>0$

If $x>1$ then $log_a(x)>0$ Well I thought that log with base a of 1 is 0. I don't know what to do more.