Questions related to real and complex logarithms.

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-5
votes
2answers
37 views

Express the following in terms of a, b, and c [on hold]

Let $\ln2=a$, $\ln3=b$, and $\ln7=c$. Express the following in terms of $a$,$b$, and $c$. A) $\ln 42$ B) $\ln 441$ C) $\ln 3.5$ D) $\ln \frac{49}{81}$
-2
votes
1answer
64 views

A question in interview for trinity college, Cambridge

Let $M$ be a large real number. Explain why there must be exactly one root $w$ of the equation $ Mx=e^x$ with $w>1$. Why is log $M$ a reasonable approximation to $w$? Write $w = \log M +y$. ...
8
votes
1answer
98 views

For which complex $a, b, c$ does $(a^b)^c=a^{bc}$ hold?

Wolfram Mathematica simplifies $(a^b)^c$ to $a^{bc}$ only for positive real $a, b$ and $c$. See W|A output. I've previously been struggling to understand why does $\dfrac{\log(a^b)}{\log(a)}=b$ and ...
-5
votes
1answer
51 views

Solve the following equation involving logarithms for $x$ [on hold]

I am having problems understanding how to solve the given equation: $$x\ln \left(x\right)+5\ln \left(x\right)-5x-25 =0$$ Any help would be very appreciated! Cheers
1
vote
2answers
40 views

Product of logartithms equation.

Please help me whilst I do a few simple school tasks. I found this one, which is unbreakable for me. I will appreciate any help. $$\log_2{x}=\frac{4}{\log_2 x-3}$$ I moved only with the fact that ...
8
votes
4answers
209 views

Calculate $\int _0^\infty \frac{\ln x}{(x^2+1)^2}dx$

Calculate $$\int _0^\infty \dfrac{\ln x}{(x^2+1)^2}dx.$$ I am having trouble using Jordan's lemma for this kind of integral. Moreover, can I multiply it by half and evaluate $\frac{1}{2}\int_0^\infty ...
0
votes
2answers
65 views

Is my proof for this limit correct?

I want to prove that $\sqrt{2 + \sqrt{2 + \sqrt{2 + \ldots}}}$ limits to 2. Let $a_0$ = $\sqrt{2}$ $a_n$= $\sqrt{2+a_{n-1}}$. Then, proving that $\sqrt{2 + \sqrt{2 + \sqrt{2 + \ldots}}}$ limits to ...
-4
votes
1answer
26 views

Evaluate $\log 64$ using the change of base formula? [on hold]

Is that even possible? I mean, there is no base.
1
vote
2answers
33 views

$\ln $ and Taylor Series Expansion (what went wrong)

Edited Problem I'm trying to express $\ln{(1-(\frac{N}{K})^{\frac{1}{4}})}$ in terms of $\ln N$, where $K$ is a constant and $1 \leq N \leq K$. This also implies $\frac{N}{K} \leq 1$. Anyone ...
1
vote
3answers
149 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 ...
1
vote
4answers
43 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: $\int \frac{1}{1-x^2}dx = \ln(1-x^2)$
4
votes
2answers
36 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
votes
3answers
42 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
votes
4answers
66 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
votes
1answer
58 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 ...
1
vote
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 $3$.
1
vote
2answers
42 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 $(x^3+1)$.
1
vote
2answers
75 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 = ...
8
votes
4answers
130 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: $2^x=8$ ...
1
vote
2answers
65 views

Question about logarithmic eqations

How to solve $4x+5^x=100$? I can't find how to solve it. I can't find a way to put the $x$'s into logarithmic form.
0
votes
2answers
74 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 ...
7
votes
5answers
968 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) ...
1
vote
1answer
33 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 ...
-4
votes
1answer
50 views

Can someone help me with this logarithm? [closed]

$$\log_{{15}}\frac{2}{9}=\log_{3}x=\log_{5}\left(1-x\right)$$
1
vote
1answer
19 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 ...
0
votes
1answer
22 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 ...
1
vote
4answers
135 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?
0
votes
0answers
18 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
36 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 ...
0
votes
2answers
43 views

Integral of logarithm of exponential function

I am trying to solve this integral: $$\int \log\left(1 + \frac{1}{\pi}\exp\left(\frac{-x^2}{2a^2}\right)\right) dx$$ where $a$ is some fixed constant. The bounds of this integral are $-a$ and $a$, ...
0
votes
2answers
37 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 ...
1
vote
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 ...
0
votes
2answers
80 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 ...
3
votes
1answer
50 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 ...
-1
votes
2answers
31 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
votes
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 ...
-1
votes
1answer
37 views

Find the area of the region of the XY -plane enclosed by the given curve with logarithm [closed]

First of all let me thank one and all for helping in prev problem, and now: Find the area of the region of the XY-plane enclosed by the curve $$y=\frac{lnx}{x}$$the line $x=e$ and the $x -axis$. My ...
3
votes
3answers
56 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
votes
0answers
51 views

Discrete logarithm with large numbers

Hi I'm trying to resolve a discrete logarithm equation : y = g^x mod p Every parameter is a 512bits number. I know the value for g, y and p and I need to find out x value. Finally, I know that g is ...
1
vote
1answer
33 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.
2
votes
2answers
56 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 ...
0
votes
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:
-4
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
votes
2answers
43 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
vote
1answer
40 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?
1
vote
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 ...
1
vote
1answer
55 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 ...
0
votes
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 ${b+2}\over{b(a+3)}$