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Questions tagged [logarithms]

Questions related to real and complex logarithms.

1
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3answers
66 views

Closed form of $\int_0^\pi \ln\left(1+\sin^2(t)\right) dt$?

I attempted to evaluate this integral but seem to be getting nowhere$$I=\int_0^\pi \ln\left(1+\sin^2(t)\right) dt$$ Wolfram returns the value $I\approx 1.18266$ but was not able to provide a closed ...
0
votes
0answers
33 views

Intersection between an AP and GP (solving equation)

Given the following equation: $$1000+(N-1)200=500\cdot 1.2^N-1$$ I need to find the value for $N$. When there was no $N$ in the left-hand-side I will be able to solve the equation with logarithms. ...
6
votes
2answers
47 views

Solve $\log_2(3^x-1)=\log_3(2^x+1)$

Solve the following equation over the real number(preferably without calculus): $$\log_2(3^x-1)=\log_3(2^x+1).$$ This problem is from a math contest held where I learn; I was unable to do much at ...
0
votes
2answers
33 views

Linear functions versus Logarithmic and Exponential functions

If a function is linear, I know that this should be true: $$ {f(x_2)-f(x_1)}={f(\bar x)} $$ Where $\bar x$ is the point exactly in between $x_1$ and $x_2$. Now, I know from looking at their graphs ...
0
votes
4answers
71 views

Solving $x-8 = x^{\log_{10}(2)}$

I am trying to solve the equation $$x-8 = x^{\log_{10}(2)}$$ This is what I managed to do so far. It is also pretty easy to find out that $x=10$ is a solution and that there are no more than 2 ...
2
votes
2answers
43 views

$e^{-x}-x=0$ solution procedure

I realized that this exponential equation has to be solved using Lambert $w$ function, I also know that the result is $x= w(1)$, but I don't know how to get there. Would you mind helping me with this? ...
0
votes
0answers
19 views

Is there a distance preserving map from the metric $d(a,b) = \log(|ab| + 1)$ to a Euclidean metric?

Let $M$ be a metric space, where the point are points in the euclidean plane and $$d_M(a, b) = \log(|ab|+1)$$ where $|ab|$ is the euclidean distance from $a$ to $b$. This is a metric since $x \mapsto \...
0
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0answers
57 views

$e^{-x} - x = 0$ - Math - Help [on hold]

Please your help to solve the following equation. Appreciate it!! $$e^{-x} - x = 0$$
1
vote
2answers
26 views

Log of products and densities

By taking the log of the function: $$\prod_{t=1}^T n_t! \prod_{i=1}^{n_t} \frac{f_v[b_{i:n_t}]}{[1-Fv(r)]},$$ it is possible to end up with the following expression: $$ \log(n_t!)+\sum_{i=1}^{n_t} \...
0
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2answers
33 views

Why are these two logs the same?

I did an integral and the answer on wolfram is $\frac{1}{5} ln{\frac{3}{2}} + ln{2}$ and it's equal to 0.77424 which is == to my answer which is $\frac{3}{5}(ln3 - ln1) + \frac{2}{5}(ln8-ln6)$ Why ...
-2
votes
1answer
21 views

Why are these equations with logs equivalent?

I have two equations: $$\frac{16}{3}(ln4 - \frac{2}{3}) + \frac{4}{9}$$ and $$\frac{4}{9}(4ln(64) - 7)$$ Why are they equivalent? So I know $2log_2{8} = log_2{8^2}$$
1
vote
2answers
55 views

What this number means: $\bar{2}.767$

I faced this situation when I was solving this problem: $\log(x) = 1.233 \Rightarrow colog (x) = ?$ The right answer was: $\bar{2}.767$
0
votes
0answers
17 views

Computing the element-wise logarithm of a matrix exponential more efficiently?

Is there any known way to compute the element-wise logarithm of a matrix exponential more efficiently? Motivation: I am trying to an optimization problem (basically finding a specific Markov ...
-2
votes
1answer
31 views

Solving $\log_x(a) = b $ and $\log_a(b) = x$ for $x$

Solving for x: $\log_a(x)=b \implies x=a^b$ $\log_x(a) = b \implies x = ?$ $\log_a(b) = x \implies x = ?$
1
vote
2answers
76 views

Complicated System of Equations involving Logarithms

I am trying to solve this system of equations. I know the answer but I am struggling with the working. I need to find $x$, $y$, and $z$ in terms of $a$, $b$, and $c$. The system of equations is shown ...
0
votes
0answers
9 views

log concavity of the survival function of the maximum of random variables

Suppose $X$ and $Y$ are two independent random variables. Let $G$ and $H$ be their cummulative distribution function respectively. We know that if both $G$ and $H$ are log-concave, so is the CDF of $\...
-1
votes
2answers
28 views

What is the error of a $\ln(x + R)$? [on hold]

I am trying to calculate the error of a $\ln(x)$ function, given my parameter $x$ has an error $R$.
1
vote
4answers
39 views

Solve for x when $(\log_x (5x))(\log_7 x)=2$

I've been trying to use the change of base property but I'm not having much luck. Can anyone give me any ideas on how I should approach this problem? The answer is 49/5 Thanks.
6
votes
1answer
44 views

How do I isolate $k$ in the following equation?

I tried to isolate the unknown $k$ in the the following equation by using logarithms, but my resolution was ugly. I am trying to learn mathematics by my own (again!), because it's a beautiful subject ...
1
vote
2answers
43 views

Solving for $x$ in terms of $y$ for equations involving quadratics, logarithms, and exponentials.

For the equation: $$y^2 = \frac{\ln(1 - xe^{xy^2})}{1 - xe^{xy^2}}$$ How might one go about solving for $x$ in terms of $y$? First, I attempted solving the equation by first performing the ...
2
votes
1answer
55 views

Solve logarithmic equation $2^x = 1 + 15\log_{5}(x+1)$

$\ 2^x = 1 + 15\log_{5}(x+1)$ Is there any other way of solving this equation, except graphical?
0
votes
0answers
19 views

How to express a series expansion into a power law form?

Suppose I have a series like $P\sim A+\epsilon B\,lnM+O(\epsilon ^{2})$ and I want to express it into a power law form. The answer is $P\sim AM^{\epsilon B/A}+O(\epsilon ^{2})$. Another example. ...
3
votes
3answers
175 views

Exponential/Logarithmic equation system

Solve the following equation system over the real numbers $$\begin{cases} x(1-\log_{10}(5))=\log_{10}(11-3^y)\\ \log_{10}(35-4^x)=y\log_{10}(9) \\ \end{cases} $$ For the functions in the above ...
-1
votes
2answers
31 views

Logarithmic equations in one variable [closed]

If $\log x\cdot y^3 =m$ and $\log x^3\cdot y^2=p$, then find $\log (\frac{x^2}{y})$. Please show all the steps necessary to solve the above. Thanks in advance.
2
votes
2answers
38 views

How do I solve this logarithm equation with different bases?

How do is solve this logarithm equation? $$11 \cdot \log_3x+7 \cdot \log_7x = 13+3 \cdot \log_4x$$ I know that I have to use the change of base formula, but I still can't figure out the equation. ...
0
votes
1answer
28 views

logarithm inequality proof

I'm trying to prove the following Log($A^m$)> Log$(\frac{B^m}{A^m})$ Where $A \in \mathbb{R}$ and $B \in \mathbb{R}$. $m \in [1,\infty]$. Any hints will be appreciated
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votes
2answers
45 views

Why does $i = 0$ or $\tau = 0$ following this logic?

Following this logic, $i$ appears to equal $0$: $$e^{i\theta} = \cos(\theta) + i\sin(\theta)$$ $$e^{i\tau} = \cos(\tau) + i\sin(\tau)$$ $$e^{i\tau} = 1 + i(0)$$ $$e^{i\tau} = 1$$ $$e^{i\tau} = e^0$$ ...
0
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0answers
23 views

Logarithmic Gain control formoula

I'm trying to implement an Automatic Gain Control mechanism for my camera. I can directly control (and read) the gain in dBs (range is 0-30 dB). The camera is monochrome and uses 8 bit. I set a ...
3
votes
3answers
25 views

For distinct positive reals $A$ and $B$, neither equal to $1$, such that $\log_A B = \log_B A$, find $AB$.

Suppose $A$ and $B$ are positive real numbers for which $\log_AB=\log_BA$. If neither $A$ nor $B$ is $1$, and if $A\neq B$, find the value of $AB$. So I use the change of base theorem getting $$\...
0
votes
2answers
57 views

Can you prove that $[(x^2 + 2x + 1)\log(x) – x^2\log(x+1)]/2\log(x)\log(x+1)$ > 1 for all x > 2?

Can you prove that $\frac{(x^2 + 2x + 1)\log(x) – x^2\log(x+1)}{2\log(x)\log(x+1)} > 1$ for all $x > 2$?
0
votes
1answer
100 views

Are $\frac{1}{6 \sqrt 2}\log |\frac{\sqrt 2 +1+3x}{\sqrt 2 -1-3x}|$ and $\frac{-1}{6 \sqrt 2}\log |\frac{ 3x+1-\sqrt 2}{3x +1 +\sqrt 2}|$ equal?

Are $\frac{1}{6 \sqrt 2}\log \biggl|\frac{\sqrt 2 +1+3x}{\sqrt 2 -1-3x}\biggr|$ and $\frac{-1}{6 \sqrt 2}\log \biggl|\frac{ 3x+1-\sqrt 2}{3x +1 +\sqrt 2}\biggr|$ equal? How? I am trying to integrate $...
0
votes
4answers
48 views

Logarithms with variables

What is a, $$\log_{\sqrt 3} \sqrt[6]a,$$ If $\log_a 27=b$ Can anyone help me with this?? A tip or a hint will be super helpful too!! The answer should include b ..
2
votes
0answers
78 views

Show that $\sum\limits_{k=1}^\infty \frac{1}{2^k k^2}=\frac{\pi^2}{12}-\frac{1}{2}\log^2 2$ [duplicate]

I have tried to split the sum into two different parts where the first reduces to the well known Basel problem, but I'm not able to show that the remaining series converge to $\log^22$: $\sum\limits_{...
0
votes
2answers
32 views

Properties of logarithms and exponents

I had a question about the properties of logarithms and exponents, that I need some assistance on: If you are familiar with electrical engineering principles, you may be familiar with Shannon's ...
0
votes
0answers
25 views

If P(x)=Q(R(x)) and $P(x)=(\log_3 x)^2$ what is R(x)?

The full problem is here. I am seriously lost on how I should approach this question. Thanks in advance.
5
votes
1answer
119 views

Evaluating $\int_0^1 (\ln(x)\ln(1-x))^n dx$

Interested in evaluating: $\int_0^1 (\ln(x)\ln(1-x))^n dx,$ where $n \in \Bbb Z^+.$ I don't really know how to tackle this problem for $n > 1.$ I was able to get to this step: $\int_0^1 (\ln(x)\...
0
votes
2answers
37 views

Find Values of $\log\bigl((1-i)^4\bigr)$

Consider $\log\bigl((1-i)^4\bigr)$. If I first expand the inside, I get $$\log((1-i)^2\cdot(1-i)^2)=\log((1-2i-1)\cdot(1-2i-1))=\log(4i^2)=\log(-4)=\ln|-4|+i\operatorname{Arg}(-4)+2ki\pi=\ln(4)-i\pi+...
0
votes
3answers
46 views

A lower bound for de Polignac's formula

De Polignac's Formula has many uses, for example when calculating the number of trailing zeroes of $n!$ :$$\nu_5(n!)=\sum_{i\le\lfloor\log_5n\rfloor}\left\lfloor\frac n{5^i}\right\rfloor.$$ For the ...
0
votes
5answers
29 views

What are the exact steps to produce $k = \log_{3/2}n $ from $(\frac{2}{3})^kn = 1$?

What are the procedures to derive $k = \log_{3/2}n $ from $(\frac{2}{3})^kn = 1$? Is there a well-known formula?
0
votes
1answer
17 views

Some confusion about complex logarithm on simple connected set (In the proof of Riemann mapping theorem) .

Let $D$ denote the unit disc , and $U$ is an open simple connected subset of $D-\{0\}$ , then we can define a square root function on $U$ by $$g(z)=e^{\frac12 \log z}$$ such that $g$ is an injective ...
0
votes
0answers
39 views

Integral of intersection of a spiral and a circle

I need to prove that for a full turn of a logarithmic spiral (independent from the constant tangent of it) and a straight line on the z axis, the logarithmic spiral always covers more space-time in ...
2
votes
0answers
69 views

Find the limit $‎{\lim\limits_{n\to\infty}}(x \log(n^2+a^2) + \sum\limits_{k=1}^n \log(k^2+a^2) -‎ ‎\log((k+x)^2+a^2))‎$‎

I've been stuck with calculating the limit of the following problem. Can you help? $‎‎\displaystyle{\lim_{n\to\infty}}(x \log(n^2+a^2) + \sum_{k=1}^n\log(k^2+a^2) -‎ ‎‎\log((k+x)^2+a^2))‎$,‎ for $a&...
-2
votes
1answer
52 views

Can someone help me with this proof? Some exponential stuff [on hold]

Let $$x=10^{1/(1-\log z)}$$ and $$y=10^{1/(1-\log x)}.$$ Show that $$z= 10^{1/(1-\log y)}.$$ Don't forget the appropriate assumptions. ($\log z$ is logarithm with base $10$ and argument $z$.)
13
votes
2answers
269 views

Geometric interpretation of the Logarithm (in $\mathbb{R}$)

(Note: limited to $\mathbb{R}$) (Note: Geometric here means with straightedge and compass) Standard approaches to introducing the concept of Logarithm rely on a previous exposition of the ...
-1
votes
1answer
21 views

solve for x when a) ln(x^2 - 1) = 3 and b) when e^2x - 3e^x +2 =0

Hi I know what is the answer actually. for ln(x^2 - 1) = 3 ; x = \sqrt((1+e) (1-e+e^(2))), -\sqrt((1+e)(1-e+e^(2))) br/ And for e^2x - 3e^x + 2 = 0; x = ln(2), 0 But I don't really know how to get ...
0
votes
1answer
18 views

Entropy of a normal distribution in Bits versus Nats in book Elements of Information Theory

This should have been easy. Converting between nats and bits is a logarithmic change of base. So going from $\log$ base $e$ to base $2$, should require the denominator to have $\log_2(e)$. However in ...
-1
votes
2answers
22 views

simplify $n^{\log _{3}(2)}$

Is there a way to simplify $n^{\log _{3}(2)}$ to have a result in the form of $xn$ ?
1
vote
3answers
63 views

Using the logarithm, find $\lim_{n\rightarrow \infty} n^{1/n}$

Using the logarithm find $$\lim_{n\rightarrow \infty} n^{1/n}$$ Here is my attempt: \begin{align} \lim_{n\rightarrow \infty} n^{1/n} &= \lim_{n\rightarrow \infty} \exp(\ln(n^{1/n})) \\ &= \...
5
votes
2answers
101 views

help with solving a limit with logarithms

I am preparing for an exam and I am struggling with the following limit: $$ \lim_{x\to 0}\frac{\ln(1+x^{2018})-\ln^{2018}(1+x)}{x^{2019}}. $$ I tried the L'Hospital rule and i tried to form ...
1
vote
2answers
43 views

Prove or disprove $n^2 \log{n} = O(n^2)$

I'm having troubles grasping Big-Oh notation and complexities. How do I go about either proving or disproving this statement?