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) $\ln \frac{49}{81}$
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.
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
28 views

Stuck on Double-Variabled Logarithm when solving this Sequences and Series question

I'm able to get to an inequality for the sum of the arithmetic sequence greater than the sum of the geometric sequence, and have solved the inequality by guess and check and have verified the ...
-2
votes
1answer
66 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 ...
-4
votes
1answer
50 views

Can someone help me with this logarithm? [closed]

$$\log_{{15}}\frac{2}{9}=\log_{3}x=\log_{5}\left(1-x\right)$$
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
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: $\int \frac{1}{1-x^2}dx = \ln(1-x^2)$
5
votes
1answer
327 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
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
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
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
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
279 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
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 = ...
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 ...
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: $2^x=8$ ...
3
votes
5answers
208 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
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) ...
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 ...
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 ...
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 ...
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 ...
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 ...
2
votes
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
153 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
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 ...
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 ...
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 $a$, ...
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 ...
1
vote
1answer
73 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
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 ...
3
votes
4answers
368 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( ...
-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 ...
0
votes
2answers
87 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 ...
-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
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.