Questions related to real and complex logarithms.

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10
votes
1answer
332 views

An equivalent for $\int_0^1\left(\frac{1}{\log x}+\frac{1}{1-x}\right)^n\;dx$

Set $$ I_n :=\int_0^1\left(\frac{1}{\log x} + \frac{1}{1-x}\right)^n \:\mathrm{d}x \qquad n=1,2,3,.... $$ We have $$I_1 =\gamma, \quad I_2 =\log (2 \pi) - \frac 32, \quad I_3 = 6 \log A - ...
0
votes
0answers
27 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
votes
1answer
44 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
38 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
54 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
votes
3answers
57 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
470 views

logarithmic function between two points

I need to find the logarithmic curve between two points $$A(0,5),\quad B(180,9)$$ We know that the formula for logarithmic function is: $\;f(x) = \log(x)\,\;$so $$ 5 = \log(0),\quad 9 = \log(180)$$ ...
1
vote
1answer
48 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
3answers
47 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
35 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
vote
4answers
54 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
votes
2answers
68 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.
0
votes
2answers
154 views

simple inverse function question

Functions in the form of $y = f(x)$ describe various sorts of line. In a quadratic line, for every extra unit in $x$, then $y$ increases by roughly $2x$. A line where for every extra unit in $x$, ...
1
vote
1answer
29 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 ...
5
votes
5answers
262 views

How to evaluate $\int_0^1 \frac{\ln(x+1)}{x^2+1} dx$

This problem appears at the end of Trig substitution section of Calculus by Larson. I tried using trig substitution but it was a bootless attempt $$\int_0^1 \frac{\ln(x+1)}{x^2+1} dx$$
0
votes
1answer
13 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
93 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
22 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
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
vote
1answer
36 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
268 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 ...
4
votes
1answer
48 views

Integration of Exponential and Logarithms, $\int_{z-1}^z \log(\frac{1}{z-y}) \exp (-| y| ^{3}) \, dy$

The integral I am dealing with is: $$\frac{3}{2 \Gamma \left(\frac{1}{3}\right)}\int_{z-1}^z \log \left(\frac{1}{z-y}\right) \exp \left(-\left| y\right| ^{3}\right) \, dy$$ where $z\in \mathbb{R}$ ...
1
vote
1answer
34 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
vote
0answers
39 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
70 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
vote
1answer
41 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
vote
1answer
25 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
2answers
58 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
vote
1answer
32 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
0answers
24 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
1answer
23 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 ...
1
vote
3answers
61 views

exact roots of $e^{ax}-x=0$

How can I find the general solution to (not a numerical approximation) $e^{ax}-x=0$ as a function of $a$. I thought maybe something like $\frac{ln(x)}{a}$.
0
votes
1answer
46 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
30 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
vote
3answers
41 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?
20
votes
1answer
221 views

$\log_2 13$ is irrational

Is it true that $\log_2 13$ is irrational? Let $x=\log_2 13\implies 2^x=13$. So, it will be an irrational number, if not,$$x=\frac p q$$ and $$2^{\frac p q}=13$$ $$\implies 2^p=13^{q}$$ Since, ...
0
votes
1answer
23 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, ...
2
votes
2answers
72 views

Does there exist any positive integer $n$ such that $e^n$ is an integer (to show $\log 2$ is irrational)?

Does there exist any positive integer $n$ such that $e^n$ is an integer ? I was in particular trying to prove $\log 2$ is irrational; now if it is rational, then there are relatively prime ...
1
vote
5answers
98 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
23 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 ...
10
votes
4answers
187 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 ...
3
votes
1answer
42 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
3answers
25k views

How to calculate anti-log using calculator?

I have a calculator that does not have antilog function. All it has is log to base 10 and natural log functions. I was wondering if it is possible to calculate antilog using the log to base 10 ...
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
votes
1answer
26 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 ...
0
votes
2answers
38 views

Even and odd functions | logarithm [closed]

show why this logarithm is an odd function? $$y = \log_2 (x-\sqrt{1-x^2})$$
3
votes
1answer
58 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 ...
-4
votes
3answers
73 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?
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 ...
1
vote
0answers
38 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 ...