Questions related to real and complex logarithms.

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1answer
47 views

Which “approximate” value of f(0.98) is this question looking for?

In a section of a calculus workbook dealing with local linearity and linear approximations of functions, the following question is posed: Consider the function f(x) = aln(x+2). Given that f'(1) = ...
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2answers
39 views

How to calculate $\log(x) = 1/2\log(16) - 1/3\log(8) + 1$

This is my first question. This is basic math, but what I get does not match the alternatives I have. So I was wondering if I did something wrong. Step 1: $\log(x) = 1/2\log(16) - 1/3\log(8) + 1$ ...
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1answer
33 views

Solving logarthmic inequality

Question Find integral solutions of this inequality$$\left (\frac{1}{10}\right )^{\log_{x-3}^{x^2-4x+3}} \ge 1$$ My try : I took log on both sides and got $\log_{x-3}^{x-1} \le-1$ but ...
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2answers
22 views

Equation solving involving logaritm

i need help solving this equation to find the variable RC: Vc=Vin(1-e^-t/RC) I already know Vc, Vin and t. I always get it wrong so the RC is negative. It represent time so it shouldn't be. Thanks! ...
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0answers
54 views

Integrals of the type $f'(z)/f(z)$

I am having trouble understanding integrals of the form: $$\int_\gamma\frac{f'(z)}{f(z)}\,{\rm d}z$$I am aware that there are problems with the complex logarithm, and we have the formula: ...
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1answer
32 views

How do I find a point on a graph which is equal on both the axis?

I have the equation $ 10^{x-0.7711} = x $. In order to find x, I thought that I'll graph the equation, and the point where x = y, will be the answer. How do I do this? Or is there any other way to ...
2
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2answers
67 views

HowTo solve this integral involving logarithm

I would like to solve integrals of the form $$I(c) := \int_0^\infty \log(1+x) x^{-c} \, dx ,$$ where $c \in (1,2)$. Mathematica says either 1) $I(c) = \frac{\pi}{1-c} \csc(\pi c)$ or 2) $I(c) = ...
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3answers
129 views

How to solve a system of logarithmic equations?

I need to create a function with the following properties: $$f(1)=1$$ $$f(65)=75$$ $$f(100)=100$$ Additionally, the function needs to grow logarithmically. So that gives three equations: $$A \cdot ...
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2answers
93 views

$\ln{\left(\frac{1}{0}\right)} = -\infty$?

I have shown it using a theorem that I made, but I am not sure, as $\lim_{\alpha \to 0^{-}}{\left(\frac{1}{\alpha}\right)} = -\infty$, and $\lim_{\alpha \to 0^{+}}{\left(\frac{1}{\alpha}\right)} = ...
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3answers
55 views

Logarithm problem with two bases

Given $$\log_x9+\log_9x=\dfrac{10}3.$$ How can I find the greatest value of $x$ that satisfies the equation above?
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1answer
59 views

Describe the Riemann surface:

$$W = \sqrt{1-z^2}$$ I would like hints only. Using @Dr.MV's hint, I get two factors: the first is $$\sqrt{(x-1)+y^2}^{\frac{1}{2}}e^{i\frac{\theta}{2}}$$, which, when we let theta range from 0 to ...
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2answers
69 views

Is the following equality true? : $\log\left(-\left(\frac{x+1}{x-1}\right)^3\right)=3\log\left(\frac{1+x}{1-x}\right)$

Is the equality below true over all complex numbers? $$\log\left(\frac{1+\frac{3x+x^3}{1+3x^2}}{1-\frac{3x+x^3}{1+3x^2}}\right)=3\log\left(\frac{1+x}{1-x}\right)$$ The L.H.S. (Left hand side ...
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1answer
24 views

Differential equations logarithmus rule question

I'm trying to understand a exercise about differential equations $x'=\frac{1}{2}x+1$ I'm going for the general solution by using separable equation. Everything goes well until I get off the rails: ...
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2answers
48 views

Solving the Logarithmic equation $\log_x (3-2\sqrt2)=2$

$$\log_x (3-2\sqrt2)=2$$ I can't solve it, I tried everything but I can't find the solution I tried logarithmic properties but nothing works, please help!
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4answers
68 views

How to compute $f(x)=x\ln{x}$ at $x=0$?

Given $f(x)=x\ln{x}$, is it correct to say that $f(0)= \lim{}_{x\to{0}}f(x)=\lim{}_{x\to{0}}x\ln{x}=0$? If this is true, why is this so? Is it because $0*n$, where $n$ is any value, is $0$? My ...
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2answers
82 views

Solving for $x$ : $a^x+b^x=c$

Well the question is to solve for $x$ in $$a^x+b^x=c \tag{a,b,c are constants}$$ Well as of me, I tried to put $\ln{}$ on both sides which does not seem to help. Apart from this I don't seem to ...
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0answers
24 views

Contour integration of logarithm: $g(\omega) \log[1 - \chi(q,\omega)]$

I'm trying to calculate the integral $$ \frac{1}{2\pi i} \int_\mathcal{C} g(\omega) \log[1 - \chi(q,\omega)], $$ where $g(\omega) = (e^{\beta \omega}-1)^{-1}$ has an infinite number of evenly spaced ...
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4answers
78 views

What is the solution to the equation $9^x - 6^x - 2\cdot 4^x = 0 $?

I want to solve: $$9^x - 6^x - 2\cdot 4^x = 0 $$ I was able to get to the equation below by substituting $a$ for $3^x$ and $b$ for $2^x$: $$ a^2 - ab - 2b^2 = 0 $$ And then I tried \begin{align}x ...
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1answer
29 views

Finding branches of $z^{ab}$

Let $f: G \to \bf{C}$ and $g: G\to \bf{C}$ be branches of $z^a$ and $z^b$ respectively ($a,b\in \Bbb C$). Suppose that $f(G) \subset G$ and $g(G) \subset G$. Prove that both $f\circ g$ and $g \circ ...
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2answers
131 views

When does $i^x=x$

Can someone please help me solve $i^x=x$? So far I have: $$i^x=x$$ $$\frac{\ln(x)}{\ln(i)}=x$$ $$e^{i\pi}=-1$$ $$e^{i\pi/2}=i$$ $$\frac{\ln(x)}{\frac{i\pi}{2}}=x$$ $$\ln(x)=\frac{i x \pi}{2}$$ ...
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4answers
81 views

Is $\log^22^k = (\log2^k)^2$?

Does the square of $\log^22^k$ include whole $2^k$? Is $\log^22^k = (\log2^k)^2 = \log2^k \cdot \log2^k = k^2$? The base of my $\log$ is $2$.
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4answers
32 views

how to solve inequality using logarithm

I was given the following expression :$$(0.87)^n\leq 0.1$$ And the next step was: $$n\geq \frac{log(0.1)}{log(0.87)}$$ What was the steps betweens?
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2answers
56 views

Showing $\lg(n!) = \Omega(n\lg n)$

I saw this equation in "Introduction to Algorithm" 3th edition in page 58 : $$\lg(n!) = \Theta(n\lg(n))$$ If $\lg(n!) = \Omega(n\lg(n))$ and $\lg(n!) = O(n\lg(n))$ then we can prove that. I can ...
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2answers
17 views

Manipulating logarithms solving probability problem

I have this equation $$(1-P_x) = (1-P_y)^{127} + 127P_y(1-P_y)^{126}$$ now I have $P_y=0.125*10^{-3}$ I've tried to solve $P_x$ using logarithms but I'm doing something wrong since $P_x$ cames out ...
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1answer
52 views

Compare between $\ln(2)$, $\ln(-2)$ [duplicate]

$x^2 =(-x)^2, \;\forall x \in \mathbb{R}^+$ $$\begin{align*} \therefore \ln(x^2) &=\ln(-x)^2\\ 2\ln(x)&=2\ln(-x)\\ \ln(x)&=\ln(-x) \end{align*}$$ If the statement above is correct, then ...
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2answers
30 views

Differentiating product and squares of logarithms

I need help differentiating. I am really confused how to solve with the $\ln x$ in the equation. Which of the logarithm rules do I need to use for this equation? $$y= 12x \ln x + 12x - 6x (\ln x)^2 + ...
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1answer
78 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 $w = \log M +y$. ...
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1answer
150 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 ...
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2answers
48 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 ...
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5answers
286 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 ...
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2answers
67 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 ...
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1answer
29 views

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

Is that even possible? I mean, there is no base.
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2answers
37 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 ...
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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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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 ...
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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 ...
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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?
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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
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 $3$.
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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
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 = ...
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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: $2^x=8$ ...
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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.
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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 ...
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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) ...
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 ...
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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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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 ...
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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?