Questions related to real and complex logarithms.

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3answers
154 views

What are the products of real solutions of this equation?

How can I solve $\:\: \log^2_{1/2}(4x)+\log_2\hspace{-0.06 in}\left(\hspace{-0.06 in}\frac{x^2}{8}\hspace{-0.06 in}\right)=8 \;$ ? I have tried the elementary for logarithms simplifying the terms in ...
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4answers
45 views

When can I use the natural log to help solve an integral?

Why is it okay to do this: $\int \frac{1}{x-2}dx = \ln(x-2)$ but not this: $\int \frac{1}{1-x^2}dx = \ln(1-x^2)$
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1answer
328 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 ...
4
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2answers
40 views

Solve for $x$ in: $e^{2\ln(x)-\ln(x^2+x-3)} = 1$

So the question is to solve for x in: $$e^{[2\ln(x)-\ln(x^2+x-3)]} = 1$$ I took the natural log of both sides, and simplified. Here is what I've gotten it down to: $$2\ln(x) = \ln(x^2+x-3)$$ And I'm ...
0
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3answers
47 views

Express the quantity as a single logarithm: $ \frac 13 \ln (x+2)^3 + \frac 12[ \ln x - \ln (x^2+3x+ 2)^2]$

Express the quantity as a single logarithm: $ \frac 13 \ln (x+2)^3 + \frac 12[ \ln x - \ln (x^2+3x+ 2)^2]$ The answer in the book is ln $\frac {\sqrt{x}}{x+1}$ If am not allowed to to cancel terms ...
2
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4answers
69 views

Prove the inequality $e^x \geq x^e$ for $x > 0$ [duplicate]

Prove that $e^x \ge x^e$ for $x \gt 0$ I applied the natural logarithm to simplify the function and I get $$\frac{x}{\ln x}\ge e$$ How to solve these types of problems?
2
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1answer
60 views

Using the complex logarithm as a conformal mapping,

I want to map the upper half plane, y>0, conformally onto the semi-infinite strip u>0, $-\pi < v < \pi$ in the w-plane. I then studied the complex logarithm, and noticed that the principal ...
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2answers
44 views

How to get the results of this logarithmic equation?

How to solve this for $x$: $$\log_x(x^3+1)\cdot\log_{x+1}(x)>2$$ I have tried to get the same exponent by getting the second multiplier to reciprocal and tried to simplify $(x^3+1)$.
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2answers
58 views

Sum of solutions of this exponential equations

How to solve this : $$x^{3-\log_{10}(x/3)}=900$$ I tried log on both sides and got nothing with exponent of $x$ and $3$.
3
votes
5answers
288 views

Calculating $\ln(1+\sqrt3)$

I distributed the natural logarithm and got $(0 + 0.549)$ [placing the values in a calculator]. However, the answer key states that the answer is $1.0051$. Where did I go wrong?
1
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2answers
96 views

How do I evaluate this integral $I = \int_{0}^{2 \pi} \ln (\sin x +\sqrt{1+\sin^2 x}) dx$?

I used some variables change to evaluate this integral but i'm not succeed may I have some wrong step as trigono-transformation.Then Is there some one who can show me how do evaluate this : $$I = ...
0
votes
1answer
23 views

Do equal rational integrands imply equal integrals, save for a constant?

Specifically, when integrating $\frac{1}{ax+b}$ we get $\frac{1}{a}\ln|ax+b|$. However, if we multiply the integrand by say $c/c = 1$, then the integral computes to $(1/a)\ln|c(ax+b)|$. Can ...
8
votes
4answers
133 views

What is the inverse of $2^x$? [duplicate]

Note: This may not be correct mathematical term, so in case of confusion, I mean what division is to multiplication. If not, just poke me in the comments. I was given this the other day: $2^x=8$ ...
3
votes
4answers
215 views

Solve this logarithmic equation: $2^{2-\ln x}+2^{2+\ln x}=8$

Solve this logarithmic equation: $2^{2-\ln x}+2^{2+\ln x}=8$. I thought to write $$\dfrac{2^2}{2^{\ln(x)}} + 2^2 \cdot 2^{\ln(x)} = 2^3 \implies \dfrac{2^2 + 2^2 \cdot ...
7
votes
5answers
1k views

Proof of the derivative of ln(x)

I'm trying to prove that $\frac{\mathrm{d} }{\mathrm{d} x}\ln x = \frac{1}{x}$. Here's what I've got so far: $$ \begin{align} \frac{\mathrm{d}}{\mathrm{d} x}\ln x &= \lim_{h\to0} \frac{\ln(x + h) ...
0
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2answers
80 views

Integration problem: $\int \ln\left(\sin(\sqrt{x})+\cos(\sqrt{x})\right)dx $

I need help in solving the following problem: $$\int \ln\left(\sin(\sqrt{x})+\cos(\sqrt{x})\right)dx $$ I really don't know how to start solving this problem; any tips or solutions will be greatly ...
2
votes
1answer
51 views

Is the expectation of log-concave function still log-concave?

I know the expectation preserves the concavity (or convexity), but I was wondering is it still true that the expectation of log-concave function still log-concave; to be more precise, Let ...
2
votes
2answers
61 views

Evaluate the limit: $ \lim_{x\to -1}\frac{x\ln(x+3) + \ln(2)} {x+1} $

$$ \lim_{x\to -1}\frac{x\ln(x+3) + \ln(2)} {x+1} $$ I tried to separate the fraction and also a change of variable (x+3 = y+1) but I couldn't solve it. Maybe there's a trivial step that I'm just ...
1
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1answer
20 views

Log, find the following values in term of m and n

I have a hard time on this log question, can you explain it? Given log(x)p = m and log(x)q = n find the following values in ...
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4answers
142 views

Is Wolfram Alpha calculating this incorrectly?

I entered in "does $2ln(x)$ equal $ln(x^2)$" into Wolfram and it came out false. Purplemath.com says that $log_b(m^n) = n · log_b(m)$. Which is correct? And why is there a difference?
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0answers
20 views

Computing the Log-Euclidean distance efficiently by using eigen-analysis.

Let $A,B\in\Bbb{S}_{++}^n$ be two symmetric positive definite $n\times n$ matrices with real entries. The Log-Euclidean distance between these matrices is defined as follows $$ d = \lVert \log(A) - ...
2
votes
1answer
37 views

$\log (A + \delta A) = ?$ (as an expansion in $\delta A$), where $A$ and $\delta A $ are matrices

$A$ and $\delta A$ are two non-commuting matrices and I am seeking a power series expansion to 2nd order in $\delta A$. After writing it as $\log (A (1 + A^{-1}\delta A) )$, I am unable to figure out ...
2
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3answers
43 views

Prove that for any positive integer $n$ and $d$, $\sum_{k=0}^d 2^k\log_2(\frac{n}{2^k})=2^{d+1}\log_2(\frac{n}{2^{d-1}})-2-\log_2{n}$

I could prove it by induction, but I need to see how I might have discovered it for myself (cause that's what's gonna be on exam).
3
votes
3answers
156 views

How to solve:$\int_0^{\infty} \frac{\log(x+\frac{1}{x})}{1+x^2}dx$

Here is my question $$\int_0^{\infty} \frac{\log(x+\frac{1}{x})}{1+x^2}dx$$ I have tried it by substituting $x$ = $\frac{1}{t}$. I got the answer $0$ but the correct answer is $\pi log(2)$. Any ...
0
votes
2answers
39 views

Maximum number of digits in numbers between 0 to $n^2-1$ of base n

The number of digits in numbers between 0 and $n^2-1$ of base n is obtained by $\log_n(n^2) = 2\log_nn = 2$ But why log is ...
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2answers
83 views

Find x, if $ \log _{15}\left(\frac{2}{9}\right)^{\:}=\log _3\left(x\right)=\log _5\left(1-x\right) $

So how can I find the value of x, if: $$ \log _{15}\left(\frac{2}{9}\right)^{\:}=\log _3\left(x\right)=\log _5\left(1-x\right) $$ I tried switching everything to base 15, but that didn't work out ...
1
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1answer
35 views

Integral formula involving logarithms and the zeros of a holomorphic function

I have the following formula I´d like to prove: Given a holomorphic function $f:U\to \mathbb C$ such that $\overline{D_r(0)}\subset U$, $f(0)\neq 0$ and $f(z)\neq 0$ for $z\in \partial D_r(0)$, we ...
1
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1answer
74 views

Homework help to rearrange formula

Given the equation $${V_m} = u(\ln {m_0} - \ln {m_8}) - g{t_f}$$ I need to solve for ${m_0}$ Here is what I have but it looks messy and I feel like there is sometihng wrong or a better way 1st ...
3
votes
1answer
53 views

How to find the center of a log spiral?

Given just a few points on a log spiral, how to find the center? Considering that the circle is a degenerate case of the log spiral, is there a way to generalize the method for finding circle centers ...
3
votes
4answers
370 views

Solve $2^{x}=x^{2}$

I've been asked to solve this and I've tried a few things but I have trouble eliminating x. I first tried taking the natural log: $x\ln \left( 2\right) =2\ln \left( x\right) $ $\dfrac {\ln \left( ...
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2answers
34 views

Log of negative numbers

I know that log of negative numbers is complex numbers. But I just got over this little proof and wondering what is wrong with this? ...
2
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2answers
25 views

$ 2\log ^2_{4}(|x+1|)+\log_4(|x^2-1|)+\log_{\frac{1}{4}}(|x-1|)=0$

Find the sum of solutions to: $$ 2\log^2_{4}(|x+1|)+\log_4(|x^2-1|)+\log_{\frac{1}{4}}(|x-1|)=0 $$ I'm not sure about what to do with the absolute values, how can I get rid of them? Should I solve ...
0
votes
2answers
90 views

Integration of log(sinx)

In trying to integrate log(sinx) and I ended up looking for a solution and found the one at this link: http://www.meritnation.com/ask-answer/question/how-to-integrate-f-log-sin-x-dx/math/766517 ...
3
votes
3answers
57 views

How to prove that $a^{\log_cb}=b^{\log_ca}$

I've met a question whereby it asked me to show that $a^{\log_cb}=b^{\log_ca}$. I'm okay with the other logarithm questions. But I don't know how to show this question out. Can anyone give some hints ...
1
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1answer
35 views

If log8n=1/2p, log22n=q, and q-p=4, find n [duplicate]

I'm having a hard time finding the value of $a$ in this problem. My teacher was trying to explain to me the process in which to get it but I did not understand him.
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votes
3answers
57 views

How to prove that: $\log_{{1\over 2}}(3) + \log_3\left({1 \over 2}\right) < -2$ [closed]

Prove that: $$\log_{{1\over 2}}(3) + \log_3\left({1 \over 2}\right) < -2$$ Please help me solve it.
0
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2answers
53 views

Where is my mistake in a logarithm?

Prove that $$3^{\log_2 5} = 5^{\log_2 3}$$ is true. Here is my solution:
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2answers
44 views

Product of all real solutions of equation $\frac {2013x}{2014}=2013^{\log_x2014}$?

How am I even supposed to start this task, i need some hint? I logarithm both sides and these are my steps: $$\frac{2013x}{2014}=2013^{\log_x2014}$$ ...
1
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1answer
59 views

Problem with this challenging summation

I'm having some trouble finding the summation of this series. I tried all I could, but in the end the denominator is creating problem. $$ \sum_{r=0}^{n} (-1)^r ...
1
vote
1answer
42 views

Finding the limit of $\lim_{n\to\infty} \frac{n^{log(n)}}{(\log n)^n}$

I try to calculate the following limit: $$\lim_{n\to\infty} \frac{n^{\log(n)}}{(\log n)^n}$$ I tried this: $$ \lim_{n\to\infty} e^{(\log(n))^2 - n \log(\log(n))} $$ Is this useful? & what ...
0
votes
2answers
26 views

If $x^y = y^x$ $(x,y \in R, x,y>0,x\neq 0 )$ and $x^p = y^q$ $(p,q \in R/\{0\}, p \neq q)$, then product $xy$ is equal to?

Solution for this one is $({\frac{p}{q}})^{\frac {p+q}{p-q}}$ , but I do not understand how I am supposed to get here, I guess something with logarithms but not sure what?
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2answers
52 views

Why is the derivative of $\log ax$, where a is any positive integer, the same?

For a question in my textbook: Differentiate $\log(2x)$ The differentiation rule for logarithm is $1/x \ln b$, where $b$ is the base. So my answer was $1/(2x) \ln 10$, but the answer my textbook ...
2
votes
4answers
40 views

Find the value of $x$ such that $(3-\log_3x)\log _{3x}3=1$.

Find the value of $x$ such that $(3-\log_3x)\log _{3x}3=1$. Is there another way to solve other than this attempt? My attempt, $(3-\log_3x)\log _{3x}3=1$ $\frac{\log(3)\left(3-\frac{\log (x)}{\log ...
0
votes
2answers
39 views

All real number solutions of equation $\log_{2011}(2010x) = \log_{2010}(2011x)$ are in certain interval. Which one is it?

This task has to be done with no calculator, but I don't have basic idea how to start. Can someone give me advice, I know this is pretty easy but I need direction for particularly this one? EDIT: I do ...
0
votes
2answers
29 views

Comparing the greatest values of two functions (Derivatives)

I've tried doing this task, and for this kind of task I should be using derivatives. When I done all the calculus, everything I got were some weird result which I do not know how to compare. Task ...
15
votes
5answers
3k views

Understanding imaginary exponents

Greetings! I am trying to understand what it means to have an imaginary number in an exponent. What does $x^{i}$ where $x$ is real mean? I've read a few pages on this issue, and they all seem to ...
0
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2answers
37 views

If $\log_23 = a$ and $\log_52=b$ then $\log_{24}50$ is equal to?

I guess this has to be done by using simple logarithmic rules, but I do not how to start. Answer in my booklet is ${b+2}\over{b(a+3)}$
0
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2answers
43 views

What is the sum of $n$ terms in the series

What is the sum of $n$ terms in the series : $$\log m+\log \left(\dfrac{m^2}{n}\right) +\log \left(\dfrac{m^4}{n^3}\right) +\cdots\cdots$$ Options $ a.)\ \log ...
1
vote
1answer
29 views

I would like to know how to do log transformation of hyperparameters in Gaussian Process Classification.

I am using Gaussian Process classification and I want to do log transform of the hyperparameters so that they are all positive. From this www.lce.hut.fi/research/mm/gpstuff/GPstuffDoc.pdf document, I ...
3
votes
4answers
148 views

How to compute the derivative of $\sqrt{x}^{\sqrt{x}}$?

I know have the final answer and know I need to use the natural log but I'm confused about why that is. Could someone walk through it step by step?