Questions related to real and complex logarithms.

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1
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0answers
34 views

(Numerical) Integration in log space

I have some function $f(x)$, which I'd like to integrate to find, $F(r) = \int_r^\infty f(x) dx $. Is there a way to do this using the values parametrized in log-space? I.e. some function $G(r, ...
2
votes
2answers
29 views

Positive entropy of system of mixed substance (mathematical viewopoint)

If two substances respectively having mass $m_1$ and $m_1$, constant pressure specific heat capacities $c_{p1}$ and $c_{p2}$, and temperatures $T_1$ and $T_1$ are mixed at constant pressure, reaching ...
1
vote
0answers
53 views

Linear to logarithm scale

I need to convert a linear range [0.1 ... 4 ... 10] to logarithmic one [0.5 ... 1 ... 5], so that 0.1 -> 0.5, 4 -> 1, 10 -> 5 I've found the similar problem here From half to double, linear to ...
1
vote
1answer
31 views

Rewriting approximated terms

The following data are inferred from a presentation slide, so I do not much info. Using linear approximation and log rules $\sqrt x $ can be rewritten as $\frac{x+1}{2}$, where $(1 \leq x \lt 2) $ ...
4
votes
1answer
214 views

How to determine periodicity of complex log in different bases?

How do you determine the "period" of a complex logarithm as a multivalued function in an arbitrary (real or complex) base? I apologize in advance if my terminology is incorrect, but let me illustrate ...
2
votes
1answer
45 views

Entropy of $f(x)=1$

Let $f(x)$ be a probability mass function $f(x) = 1$ on $x = [0,1]$, and entropy defined as $$H(p(x)) = -\int p(x) \log_2(p(x)) \, dx$$ where $p(x)$ is a pmf. Unless I've made an arithmetic error, the ...
3
votes
1answer
25 views

Solving equations with logarithmic exponent

I need to solve the equation : $\ln(x+2)+\ln(5)=\lg(2x+8)$ With the change of base formula we can turn this into: $\ln(x+2)+\ln(5)=\frac{\ln(2x+8)}{\ln(10)}$ We can also simplify the LHS with the ...
4
votes
7answers
135 views

How do i convince students in high school for which this equation: $2^x=4x$ have only one solution in integers that is $x=4$?

I would like to convince my student in high school level using a simple mathematical way to solve this equation: $$2^x=4x$$ in $\mathbb{z}$ which have only one integer solution that is $x=4$ . ...
2
votes
5answers
135 views

Nice algebra question

The real numbers $x,y,z$ satisfy $$x+y+z=4$$ $$xy+yz+zx=2$$ $$xyz=1$$ Then $x^{3}+y^{3}+z^{3}$=??. It's for sharing a new ideas, thanks:)
2
votes
1answer
70 views

When is ${\log(a) \over \log(b)}$ an integer?

I've encountered this quite a bit. If I have ${\log(a)\over \log(b)} = c$ where $b$ is a known positive integer, what can be said about $a$ if $c$ needs to be an integer?
8
votes
3answers
12k views

Why must the base of a logarithm be a positive real number not equal to 1?

Why must the base of a logarithm be a positive real number not equal to 1? and why must $x$ be positive? Thanks.
25
votes
2answers
565 views

A difficult logarithmic integral ${\Large\int}_0^1\log(x)\,\log(2+x)\,\log(1+x)\,\log\left(1+x^{-1}\right)dx$

A friend of mine shared this problem with me. As he was told, this integral can be evaluated in a closed form (the result may involve polylogarithms). Despite all our efforts, so far we have not ...
-1
votes
1answer
57 views

Inequality with two variable [closed]

Proof this inequalities: $$\frac{x-y}{x} < \ln\frac{x}{y} < \frac{x-y}{y}, $$ if $0<x<y$. $$\frac{b-a}{\cos^2a} < \tan b-\tan a <\frac{b-a}{\cos^2b},$$ if $0<a<b<\pi/2$.
0
votes
3answers
45 views

Inequality $\left(\frac {17}{25}\right)^k \le 10^{-5}$ - Solve for $k$

How can I solve for $k$ the following inequality : $$ \left(\frac {17}{25}\right)^k \le 10^{-5} $$ This is what I got so far. By taking $\log_k$ from both sides I get: $$ \log_k{\left(\frac ...
9
votes
2answers
187 views

Checking logarithm inequality.

Which one of the following is true. $(a.)\ \log_{17} 298=\log_{19} 375 \quad \quad \quad \quad (b.)\ \log_{17} 298<\log_{19} 375\\ (c.)\ \log_{17} 298>\log_{19} 375 \quad \quad ...
1
vote
2answers
26 views

Branch points and natural maximal domain of $\log\frac{1+z}{1-z}$

So I have a function $\log\frac{1+z}{1-z}$ and I'm supposed to find its branch points, natural maximal domain. So far I converted $z$ to $x+iy$ but no real good. Can't check holomorphicity using ...
0
votes
0answers
10 views

Pseudo-log transformation returning positive values

I'm looking for a transformation that acts similarly to natural log, but I want to return positive values only. My untransformed values range from 0.01 to 50. Of course I could simply offset the log ...
1
vote
1answer
51 views

For which value(s) of $b$, are $b\ln(x)$ and $b^{x} $tangent to each other?

For which positive value(s) of $b$, are $b\ln(x)$ and $b^{x} $tangent to each other? By setting the derivatives of the two curves equal to each other: $$\dfrac{b}{x}=b^{x}\cdot \ln(b)$$ ...
2
votes
4answers
67 views

Derivative of $e^{\sin(\ln[2 \arctan(2x)])}$ [closed]

Let $$F(x) = e^{\sin(\ln[2 \arctan(2x)])}$$ and take the first derivative of $F(x)$. Could someone walk me through each step here? It seems to be a pretty straightforward chain rule problem but it ...
2
votes
3answers
46 views

Minus sign in logarithm of integral's solution

I want to solve the following integral: $ \int \frac{dp}{a(1-p)u-bp} $ where $a$, $b$ and $u$ are some constants. After integration I get: $ p = -\frac{\log(-apu+au-bp)}{au+b} + C. $ According to ...
3
votes
2answers
31 views

Maximizing Theta in a Summation Formula

I need to take the first derivative of $$\sum Y_i (\log(\Theta )) +(n-\sum Y_i)(\log(1-\Theta )) $$ with respect to theta, and then solve for theta. I believe this is my derivative... $$\frac ...
2
votes
2answers
22 views

The number of logarithm applications to get from n below 1

Let $L(n)$ to be a number of logarithms that you need to apply on $n$ until you get below 1: $$ 0 \leq \log\cdots\log n < 1 \\ \uparrow \\ L(n)\mbox{-times} $$ Is there a name for this function? ...
6
votes
3answers
106 views

Closed-form of $\int_0^1 x^n \operatorname{li}(x^m)\,dx$

I've conjectured, that for $n\geq0$ and $m\geq1$ integers $$ \int_0^1 x^n \operatorname{li}(x^m)\,dx \stackrel{?}{=} -\frac{1}{n+1}\ln\left(\frac{m+n+1}{m}\right), $$ where $\operatorname{li}$ is the ...
0
votes
1answer
43 views

Relation between number of digits of a number and its logarithm? [duplicate]

I found a couple of questions where, for example, they ask you to calculate the number of digits in $18^{200}$ and only the value of $\log 18$ is given. Can anyone tell me a way?
1
vote
1answer
26 views

Solving natural log equations

The equation $\ln(|y+1|)= x-2$ where you solve for $y$, I am just unsure of how the absolute value plays into this. I am assuming that I would convert to exponential form to get $|y+1|=e^{x-2}$ and ...
30
votes
2answers
554 views

Closed form for $\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx$

I need to evaluate this integral: $$Q=\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx.$$ I tried it in Mathematica, but it was not able to find a closed ...
11
votes
2answers
157 views

Existence and uniqueness of a function generalizing a finite sum of powers of logarithms

I hope to find a proof of the following conjecture: $(1)$ For every $a>0$ there is a convex analytic function $f_a:\mathbb R^+\to\mathbb R$ such that: $f(1)=0$ and $\forall x>1,\ ...
3
votes
1answer
139 views

Find the number $n^{2}$ from the number $n^{n^{n^{2}}}$

Find the number $n^{2}$ from the number $\large n^{n^{n^{2}}}$ Any help? I tried with $\log$ but I got nothing.
0
votes
1answer
35 views

LN word problem

measurement of a child's ability to learn is given by the function $$L(t)=\frac{ln(t+1)}{t+1}$$ where t is the child's age in years, for $0 ≤ t ≤ 5$ At what age does a child have the greatest ...
1
vote
1answer
19 views

Logarithmically bounded function fulfills $f(n) \le \lceil m \cdot \log_b r \rceil$ for certain numbers $n,m,r$

Let $f : \mathbb N \to \mathbb N$ be a function such that $f(n) \le 1 + \log_b n$ for some base $b$ and all $n$. Now let $n \in \mathbb N$ have the property that $$ \frac{r^m - 1}{r-1} \le n < ...
2
votes
2answers
601 views

How to recognise intuitively which functions grow faster asymptotically?

There are some cases where it is not so simple to decide which function grows faster asymptotically. For example, in the following cases, why (intuitively) $g(n)$ should grow faster than $f(n)$, or ...
1
vote
2answers
48 views

( Logarithmic Equation ) Solve for x.

$(x+1)^{log(x+1)} = 100(x+1)$ Attempt at solution : $$ (x+1)^{log(x+1)} = 100(x+1)$$ $$= x^{log(x+1)} + 1 = 100x +1$$ $$=(x+1)+1=100x+1$$ $$=−98=99x$$ $$x=−98/99$$ But the answer given in the ...
4
votes
2answers
66 views

Iterative calculation of $\log x$

Suppose one is given an initial approximation of $\log x$, $y_0$, so that: $$y_0 = \log x + \epsilon \approx \log x$$ Here, all that is known about $x$ is that $x>1$. Is there a general method of ...
0
votes
0answers
21 views

What would be the equation for “3% in 100 Hz range, 0.5% in the 2000 Hz range”

The smallest distinguishable pitch/frequency by a human ear is something like this: Pitch is our perceptual interpretation of frequency. As mentioned, ideal human hearing ranges from 20 to 20,000 ...
1
vote
2answers
66 views

Why does $\lim_{x\to0^{-}} \mathrm {Im}\left( \mathrm \ln \left(x\right)e^x\right)=\pi$?

Why does $$\lim_{x\to 0^{-}} \mathrm {Im} \left( \ln\left(x\right) e^x\right)=\pi$$ Obviously this is no coincidence. I was thinking maybe this has to do with Euler's formula, but I don't see how the ...
3
votes
4answers
91 views

How to solve $\lim_{x\to1}\left[\frac{x}{x-1}-\frac{1}{\ln(x)}\right]$ without using L'Hospital's rule?

$$\lim_{x\to1}\left[\frac{x}{x-1}-\frac{1}{\ln(x)}\right]$$ How can I solve this without using the L'Hopital's rule? Any tips or hints would be greatly appreciated. I tried using the substitution ...
5
votes
1answer
336 views

inequality $10<2^{2^{\frac {3}{\log_2 \log_2 10}}}$

While working on this question I ended up with $10<2^2{^{\frac {3}{\log_2 \log_2 10}}}$ I am looking for answers using methods similar to this or this or this or this. Alternative original ...
-1
votes
2answers
29 views

What is the Solution? x ≥8lgx

x ≥ 8lgx I have to find which x satisfy this inequality. I found the points using graph, but I'd like someone to show me how to find it without it.
1
vote
1answer
42 views

Logarithm Propr

I'm having a bit of trouble proving the following property: Theorem If $Re(z)>0 $ and $ Re(w)≥0$, then $\log(zw)=\log(z)+\log(w)$, where log is the principal branch. I know that $\log (zw) = ...
1
vote
2answers
44 views

Show $n \cdot \log_b r + \log_b \frac{r}{r-1} \le \lceil n \cdot \log_b r \rceil$

Let $n$ be a natural number and $b, r > 1$ be two natural numbers, then I guess we have $$ n \cdot \log_b r + \log_b \frac{r}{r-1} \le \lceil n \cdot \log_b r \rceil. $$ where $\lceil x \rceil = ...
3
votes
3answers
199 views

Indefinite integration: $\int x^{x^2+1}(2\ln x+1)dx$

Find the value of the integral: $$\int x^{x^2+1}(2\ln x+1)dx.$$ My attempt: I tried by using integration by parts, but not working since $x^{x^2+1}$ keeps coming again and again. Then I tried putting ...
7
votes
4answers
705 views

Limit to infinity with natural logarithms

I found the following problem in my calculus book: Solve: $$\lim_{x\to \infty } \left(\frac{\ln (2 x)}{\ln (x)}\right)^{\ln (x)} $$ I tried to solve it using log rules and l'Hôpital's rule with no ...
16
votes
6answers
514 views

$\log_9 71$ or $\log_8 61$

I am trying to know which one is bigger :$$\log_9 71$$ or $$\log_8 61$$ how can i know without using a calculator ?
0
votes
1answer
27 views

Linear, Squared and Logarithmic scales with given input domain and output range

The input domain is $[12,24]$ and the output range is $[0,720]$. Now I know that with using linear scaling the value $16$ of the input range is mapped to $240$; with using sqrt scaling the same value ...
0
votes
2answers
38 views

Logarithms and Taylor Series

Before Log Tables, how were they able to compute expressions such as $2^{2.221}$? I understand they could take a Taylor expansion of $\frac{1}{x}$, but how were they able to condense the expansion ...
0
votes
1answer
22 views

Minimum of the function $b\log_b x$

Why the function $b\log_b x$ has its minimum at $b=e$? How to explain this? I'm asking because I can't understand why ternary base has more economy than binary: ...
3
votes
4answers
80 views

Solve for $x$ - Logarithm Equation $\ln x+\ln(x+1)=\ln 2$

My attempt: $\ln x(x+1)=\ln 2$ $e^{\ln x(x+1)}=e^{\ln 2}$ $x(x+1)=2$ $x^2+x-2=0$ $(x-1)(x+2)=0$ therefore $x=1, -2$
1
vote
5answers
80 views

logarithm proof fallacious or not?

$e^{-x}=e^{1/x}$ Taking the natural logarithm of both sides $$\ln(e^{-x})=\ln(e^{1/x})$$ $$-x=1/x$$ $$-x^2=1$$ $$x^2=-1$$ $$x=i$$ I know I am doing something wrong here. Also can someone please ...
0
votes
1answer
17 views

Geometric distribution with given probability value.

The probability of a man hitting a target is $2/3$. If he doesn't stop shooting until he hits the target for the first time, a) What is the probability of taking 5 shots to hit the target? b) Which is ...
-3
votes
1answer
36 views

Express as a single logarithm [closed]

Hi I need to express the following and have no clue how to do so. $$\ln(x+3)-3\ln(x-7)-\ln(x+8)$$ Can someone please help