Questions related to real and complex logarithms.

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2
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1answer
71 views

Checking derivation of y = a^x

Can you tell me if there are any flaws with this derivation of $y = a^x$... The assumptions are that the derivative $$\frac{d}{dx}e^x = e^x$$ and that the derivative $$\frac{d}{dx}\ln x = ...
0
votes
1answer
44 views

Is there a property for log(n)/n?

I found a small exercise which I couldn't figure what to do, so I found a solution. Then I tried to understand it and everything went well until I got to this part: $$\frac{1}{8} = ...
0
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2answers
51 views

Help me to solve math homework on logarithmic

How to solve this math home work? Please help.. What is the value of $\log \left(\dfrac{i\pi}{2}\right)$ ? I got to know the answer is "$\dfrac{i\pi}{2}$", but don't know how to solve it. Please ...
0
votes
1answer
31 views

Finding the exponent of $2$ such that $x \cdot 2^a$ is as close to $1$ as possible

How do I find an exponent of $2$ that when multiplied with another number would bring the result closest to the positive side $1$? Like this: $y = x \cdot 2^a$, where $y\ge 1$ has to be as small as ...
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vote
3answers
41 views

If $2 \cdot log_e{(x -2y)} = log_e{y} + log_e{x}$, then find the numerical value of $\frac{x}{y}$

If $2 \cdot log_e{(x -2y)} = log_e{y} + log_e{x}$, then find the numerical value of $\frac{x}{y}$ My try: $2 \cdot log_e{(x -2y)} = log_e{y} + log_e{x}$ $log_e{(x-2y)^2} = log_e{xy}$ ...
0
votes
1answer
27 views

Mathematics - geometric progression question

If $a$, $b$ and $c$ are in geometric progression, then what are $\log_ax$, $\log_bx$ and $\log_cx$ in? What I did: I substituted values for $x, a, b$ and $c$ and tried to solve it further. What I ...
2
votes
1answer
57 views

What does this log notation mean?

Can someone please explain what $^2\log x$ means? Is it the same as saying $\log x^2$ or is it something completely different? Here is an image of it as an example:
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1answer
37 views

Characteristic function of logarithm of random variable

If I know the characteristic function $\phi_X(t)$ of a random variable $X>0$, how can I write the characteristic function $\phi_Y(t)$ of $Y=\log(X)$? I know that $\phi_X(t)=E[e^{itX}]$ and ...
6
votes
1answer
173 views

$\exp(\ln(x))=x$ and $\ln(\exp(y))=y$.

Let $(A,1_A,|\cdot{}|)$ be a unital Banach algebra, for instance $A=M_n(\Bbb R)$ or $M_n(\Bbb C)$. What is the union of all open unit balls $B_{\|\cdot{}\|}$ where $\|\cdot{}\|$ ranges over all ...
0
votes
1answer
25 views

How to define this logarithmic function

I am trying to get my head around the definition of this function (that I concocted as an exercise in defining a function). Let $f$ denote the function satisfying: $f(0) = +\infty$, and $f(+\infty) = ...
0
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1answer
47 views

Show that $\frac 1{\log_2x}+\frac 1{\log_3x}+\cdots+\frac 1{\log_{43}x}=\frac 1{\log_{43!}x}$ [closed]

Show that $\frac 1{\log_2x}+\frac 1{\log_3x}+\cdots+\frac 1{\log_{43}x}=\frac 1{\log_{43!}x}$.I am just not able to get it.please help.
0
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2answers
22 views

definition or property of logarithms

I've seen a lot of complicated logarithm definitions on this StackExchange and I have a rather simple question: $$a^{b}=c \leftrightarrow \log_a{c}=b$$ Is this a definition of logarithms, which all ...
0
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2answers
37 views

Logorithms on a first level learning

Solve log$_{5x-1}$ $4$ $=$ $1/3$ $(5x-1)^{1/3}$=4 $((5x-1)^{1/3})^3$ = $4^3$ $5x-1=64$ $5x=65$ $13$ I am not sure where to go with this. I learned some things about logs before my class ended ...
0
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3answers
74 views

Question releating to the $\int^x_1\frac{\ln(t)}{t+1}$

If $f(x)=\int^x_1\frac{\ln(t)}{t+1}dt$ if $x > 0$. Compute $f(x) + f(1/x)$. As a check, you should obtain $f(2)+f(1/2)=(\ln2)^2$ I have tried evaluating the integral ...
2
votes
0answers
35 views

Interpolation of iterated logarithms

$$\text{Let }\log^2(x)=\log(\log(x)),\\ \text{ then }f(y,x)=\log^{\lfloor1+y\rfloor}\left(\log(x)/\log((1-x^{1/x}(y-\lfloor y\rfloor))+(y-\lfloor y\rfloor))\right)$$ gives an interpolation between ...
3
votes
3answers
81 views

calculate $\int_{0}^{\pi} \int_{0}^{x}\log(\sin(x-y))dydx$

I was asked to find the integral $\iint_A \log(\sin(x-y))dxdy$ where $A$ is the triangle $y=0, x=\pi, y=x$ in the first quadrant. I was given a hint: evaluate $\int_{0}^{\pi}\log(\sin(t))dt$ using ...
1
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1answer
45 views

$\sum_{x=a}^{b-1}\frac{1}{x}$ and $\sum_{x=a+1}^b\frac{1}{x}$

I have to prove the following relations: $\sum_{x=a}^{b-1}\frac{1}{x}\geq\log b - \log a $ $\sum_{x=a+1}^{b}\frac{1}{x}\leq\log b - \log a $ I tried to use the relation that $\int_a^b \frac{1}{x} ...
0
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3answers
32 views

Logarithm deduction question

Given that $\log_{10}2 = 0.3010$ to four decimal places and that $10^{0.2} < 2$, is it possible to deduce that: $2^{100}$ begins in a $1$ and is $30$ digits long; $2^{100}$ begins in a $2$ and is ...
0
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1answer
42 views

How to calculate arithmetic mean of log values

I am working with really small values of probabilities and that is why their log values are used. So for example, let probA and probB be some normal values of probabilities of two events and because ...
1
vote
1answer
31 views

Help with Evaluating a Logarithm

A precalculus text asks us to evaluate $\log_{8}\dfrac{\sqrt{2}\cdot\sqrt[3]{256}}{\sqrt[6]{32}}$ I do the following: $\log_{8}\dfrac{\sqrt{2}\cdot\sqrt[3]{(2^2)^3\cdot 2^2}}{\sqrt[6]{2^3\cdot 2^2}}$ ...
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3answers
51 views

Determine all positive numbers $a$ for which the curve $y = a^x$ intersects the line $y = x$ without calculus

The answer is $0 < a < e^{1/e}$ , but how to find it? Is it a system of equations? Which ones? I just need an idea at least, because I'm stuck. If it is impossible without calculus, solve it ...
0
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2answers
39 views

Avoiding substraction for finite difference with log and exp

I want to approximate the derivative of f(x) Finite difference $f'(x) \approx \frac{f(x+h)-f(x)}{h}$ I was taught that the error from the substraction is blown up for small h. This I can verify ...
0
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2answers
28 views

How to interpret the difference in log points

How can we interpret the difference between two log points? Is it correct to interpret this difference in percentage points? Thanks. Marko
0
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1answer
29 views

Best way to handle the ratio which cannot be represented as floating point numbers.

I need to calculate the ratio of the form: $s=\sum_1^3q_i$,$\quad$ $p_i=\frac{q_i}{\sum_1^3q_i}$, where $q_i >0$. One problem is that $q_i$ are too small that they can not represented as ...
2
votes
4answers
461 views

Infinite sum of logarithms

Is there any closed form for this expression $$ \sum_{n=0}^\infty\ln(n+x) $$
1
vote
1answer
50 views

Solving $F(x) = 0.3$ where $F(x) = 1-\frac{200^{ 2.5 }}{ x^{2.5}}$

Consider the function $$F(x) = 1-\frac{200^{ 2.5 }}{ x^{2.5}}.$$ We want to solve $F(x) = 0.3$. Since $ F(x) = 0.3$ then we can say, $2.5 \ln (\frac{200}{x}) = 0.7$. $\ln(\frac{200}{x}) = 0 .28$. ...
1
vote
1answer
30 views

How can I read logarithmic scale?

I've got this histograms: How can I read that logarithmic scale? For example, on the histogram 1 there is approximately $10^{-3}$ value at y-axis at 2 value at x-axis. Does it meant that there is a ...
1
vote
3answers
34 views

Find $log_{p}X^2$?

Given that $log_{p}X=5$ and $log_{p}Y=2$, find i) $log_{p}X^2$ I did this, $X=p^5$ and $Y=p^2$ But how do I use them? Should I find $p$?
5
votes
3answers
59 views

Limit of logarithmic function using l'Hospital

How can I find the following limit: $$\lim_{x\rightarrow \infty}\frac{\ln(1+\alpha x)}{\ln(\ln(1+\text{e}^{\beta x}))}$$ where $\alpha, \ \beta \in \mathbb{R}^+$. My first guess was to use ...
0
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0answers
27 views

Do so big $p \in \Bbb N : \lim_{n \to \infty} \frac{\ln^p {n} }{n} = A \ne 0, A \in \Bbb R$ exist?

We know $$\lim_{n \to \infty} \frac{\ln {n} }{n} = 0$$ $$\lim_{n \to \infty} \frac{\ln^n {n} }{n} = \lim_{n \to \infty} \frac{n\ln^{n-1} {n} }{1} = \infty$$ For usual $p \in \Bbb N $: $$\lim_{n \to ...
2
votes
1answer
128 views

What is ${\mathfrak{R}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$?

This is a new integral that I propose to evaluate in closed form: $$ {\mathfrak{R}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$$ where $\log (z)$ denotes the principal value of ...
0
votes
1answer
21 views

How to clear variable $v$ from logarithmic equation

I have the following: $6.4 = -\log\dfrac{5-v*0.1}{50+v}$ I would like to know how to solve the equation in order to get $v$'s value. Thank you very much.
2
votes
1answer
76 views

-ln(0.1) equalling to ln(10)?

I am having quite a headache wrapping my head around this solution. I do not understand the first line where they get lambda = ln(10) from statement to the left. Somebody please explain this to me. ...
0
votes
4answers
46 views

How do you solve this using only given values, logarithm rules and no calculator?

Given that $\log12=1.0792$ and $\log4=0.6021$, solve $\log8$ without a calculator. I am familiar with the following three rules: Product rule: $\log(a\cdot b)=\log a+\log b$ Quotient rule: ...
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0answers
19 views

Combining ±% with ±dB in measurement uncertainty

Firstly apologies if this is not the correct place to post this but wasn't sure which site would be good to ask regarding about measurement uncertainty calculation. I am trying to calculate the ...
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2answers
78 views

What is the best way to calculate log without a calculator?

As the title states, I need to be able to calculate logs on paper without a calculator. For example, how would I calculate $log(25)$ ?
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2answers
22 views

Canceling out log2

How would go about cancelling out the $\log_2$. Is it possible for the TI 89 to handle this? I'm not sure how to put $\log_2$ in my TI 89. $20=30\cdot \log_2(1+x)$
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2answers
60 views

Is the inverse ackermann function the slowest growing function that goes to infinity?

Actually, this is not precisely my question. If $a(x)$ is the inverse ackermann function, then obviously $a(a(x))$ grows slower than $a(x)$, as does $\log(a(x))$, and so on. But is there a function f ...
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2answers
70 views

Commutation between Logarithm and Gaussian Integral.

I'm calculating a partition function (physics) and I arrive to the following expression: $$\log \int_{-\infty}^{\infty} \frac{du}{\sqrt{2\pi}} e^{-u^2/2} e^{-nq/2}[2\cosh(\sqrt{q}\,u+m)]^n \qquad(1)$$ ...
1
vote
1answer
30 views

Domain of definition of the function

I was going through some questions of Relations and Functions and now I am stuck to one. Question says Question: Domain of definition of the function $$f(x)=\frac{9}{9-x^2}+\log_{10}(x^3-x)$$ ...
0
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2answers
87 views

A logarithm integral

Calculate the integral \begin{align} \int_{0}^{1} \frac{ \ln(\sqrt{x} - \sqrt{1-x}) }{ \sqrt{x} } \ dx \end{align} and show the value is negative.
2
votes
3answers
53 views

Determining $\lim_{n\to\infty}\left(n^{\tfrac{1}{n}}-1\right)^n$ with only elementary math

I am trying to find this limit: $$\lim_{n\to\infty}\left(n^{\tfrac{1}{n}}-1\right)^n,$$ I tried using exponential function, but I see no way at the moment. I am not allowed to use any kind of ...
3
votes
2answers
77 views

$\ln(n)/n<1/2$ proof without calculus or any kind of advanced mathematics

Is it possible to show that $\ln(n)/n<1/2$, for all natural numbers $n$ without using calculus, but just some elementary math? Induction is allowed. I was trying to show equivalently that ...
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0answers
43 views

How do I apply law of log here

My function is defined as How do I find log M(t)?
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0answers
24 views

applying logarithm law question

Here is my equation (below) on which I am applying log $X=\frac{a}{b}\left ( c-d \right )$ so far I applied it as $\log X=\log(a)-\log(b)+\left [ \log\left ( c \right )-\log\left ( d \right ) ...
1
vote
1answer
22 views

If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$.

If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$. I tried to separate the terms first and I got $\dfrac12 (\log(1+\log x) - \log(1-\log x))$. The answer is $\dfrac1{x(1-\log x)^2}$.
0
votes
1answer
24 views

An efficient technique to test if an exponential of logs gives an integer

Is there an efficient way of testing if the resulting value of an exponential gives an integer without actually expanding the equation. For example: $ {\log(12) - \log(4)}=1.09861\ldots $ and is a ...
0
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0answers
24 views

does $\ln{a}/\ln{b}=\log_ba$ still stand when $a,b\in\Bbb C$? ($b\ne1$)

I've heard that the property of logarithm becomes to have some differences with complex numbers. I'm not sure whether I should apply or not the property that's used with positive numbers.
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0answers
55 views

Logarithm and “basic” functions.

To express the antiderivatives of $\frac{1}{x}$, we cannot apply the formula $\int x^n dx=\frac{x^{n+1}}{n+1}+C$ and we need to introduce a new function, the logarithm. But how can we prove that ...
0
votes
1answer
31 views

Sketching the graph of $y =\ln(4-x)$

$y = \ln(4 - x) $ This graph has two operations applied to the $\ln x$ graph - a reflection and a translation. If you reflect the graph in the $y$-axis first, and then shift the graph 4 units to ...