Questions related to real and complex logarithms.

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Prove that $\log{\frac{\sum_{j=1}^n x_j}{n}}\ge \frac{\sum_{j=1}^n \log x_j}n$ for $j=1,…,n$

Prove that $\log{\frac{\sum_{j=1}^n x_j}{n}}\ge \frac{\sum_{j=1}^n \log x_j}n$ for $j=1,...,n$. My Work. LHS: \begin{align}\log{\frac{\sum_{j=1}^n x_j}{n}}&=\log{\sum_{j=1}^n x_j}-\log ...
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4answers
79 views

Why is this proof considered wrong? [closed]

I was asked to prove following statement: $$\log_a{(x_1\cdot x_2)} = \log_a{(x_1)} + \log_a{(x_2)}$$ What I did was: \begin{align} \log_a{(x_1\cdot x_2)} &= \log_a{(x_1)} - (-1)\cdot ...
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1answer
47 views

How to solve $\lim_{x\rightarrow \infty }\dfrac {2} {x}\sum _{k\rightarrow 1}^{x}\ln \left( \dfrac {x+k} {x}\right)$?

Let k be positive integers $\lim_{x\rightarrow \infty }\dfrac {2} {x}\sum _{k\rightarrow 1}^{x}\ln \left( \dfrac {x+k} {x}\right)$
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2answers
24 views

In Need of Logarithms Simplification Exercises

I am very interested in mathematics, however, finding nowhere near wanted information in school sometimes I go and learn something by myself. Just like this time. I decided to learn more about ...
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1answer
40 views

Find the values of $x$ satisfying the equation $ { \log_{5} }^2 x + \log_{5x} {5\over x } = 1 $?

$$ { \log_{5} }^2 x + \log_{5x} {5\over x } = 1 $$ My progress $$\begin{align} &{ \log_{5} }^2 x + \log_{5x} {5\over x } = 1 \\[2ex] \implies & { \log_{5} }^2 x + \log_{5x} {25} \cdot ...
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6answers
37 views

Logs with exponential bases

I know that $e^{\log_{e^2} (16)}$ is 4, but I can only get this far: $$e^{\log_{e^2}(4^2)}$$ $$e^{2\log_{e^2}(4)}$$ I need some way to cancel the 2's so I can get $$e^{\log_e(4)}$$ but I don't the ...
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2answers
27 views

Solve for $x$ in an system of a linear and logarithmic equation.

I have this question on my homework: $$\\f(x)=\ln{(x-3)}\\g(x)=\frac12x-7\\\text{solve for x in:}f(x)=g(x)$$ I have used the substitution property to get this: $$\ln{(x-3)}=\frac12x-7$$. I don't ...
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0answers
45 views

Solving for $x$ in $(y-x)\ln\frac{x}{y} = a$

I have the expression $$(y-x)\ln\frac{x}{y} = a,$$ and I want to express $x$ in terms on $y$ and $a$. I know that in this kind of problem, the Lambert function $W$ is likely to show-up, but that ...
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2answers
27 views

Need help with expressing this logarithm

Express $log_3(a^2 + \sqrt{b})$ in terms of m and k where $m = log_{3}a$ $k = log_{3}b$ Given this information I made $a = 3^m$ $b = 3^k$ Therefore = $log_{3} ((3^m)^2 + (3^k))^{\frac{1}{2}}$ = ...
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3answers
33 views

Solving a log equation for two variables

Goal is to find both $\beta$ and $\omega$. I already have the answer here, but I'm confused as to how to get it. $\log_6 250 - \log_\beta 2 = 3 \log_\beta \omega$ This is what I did: $\log_6 250 = ...
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0answers
46 views

Are known these identities, that I've deduce using Mobius inversion formula?

I would to know if this formula is right and know (these formula are the same by exponentiation), since I deduce this easily by a standar way (perhaps there are mistakes) using Mobius inversion from ...
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2answers
21 views

Unsure how to treat y in this derivative/log problem

Need to find the derivative of $h(y)= \ln(y^2 \cos y)$ Treating it like a normal variable like an x isn't working for me, the way we used y's in earlier problems where you get a y' in there doesn't ...
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1answer
25 views

Logarithmic differentiation trouble with bottom of fraction

$y=\frac{(2x+3)^9}{\sqrt x(x^2-x)^6}$ I switched it to $\ln(y)=9\ln(2x+3)-6x^{1/2}\ln(x^2-x)$ and then used the log rules for derivatives I know and the product rule on the right side and wound up ...
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3answers
47 views

computing $\ln[(1+i)^{2}]$

How to compute Natural logarithm of complex numbers? and how to verify our answer? in example: $\ln[(1+i)^{2}]$
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1answer
31 views

Determine the form of set containing the real solutions

From the following inequality: $\dfrac{\log_{2^{x^2+2x+1}-1}(\log_{2x^2 + 2x + 3}(x^2 - 2x)}{\log_{2^{x^2+2x+1}-1}(x^2 + 6x + 10)} \geq 0 $, the set of all real solutions to this inequality is of ...
2
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1answer
40 views

Computing the Taylor expansion of the square root of cos(z),

Let $\large f(z)=\sqrt{cosz}$ with the branch of the square root chosen so that $f(0)=1$. Consider the power series expansion of $f(z)$ in powers of $z$. Part 1) Compute the first three non-zero ...
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2answers
53 views

How to get $y(x)$ out of $y = \sqrt[3]{xa(\log_{10}(yb)-c)^2}$

I am trying to calculate the speed of a boat given the power being delivered by the motor. Unfortunately, the friction coefficient is speed dependent and is inside a $\log_{10}$. Is it possible to ...
2
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1answer
50 views

Convergence/divergence of the series $\sum_{n=4}^\infty\frac{(-1)^n}{\log(\log(n))}$?

We have the following sum: $$\displaystyle \sum_{n=4}^\infty \dfrac{(-1)^n}{\log(\log(n))} $$ I have a hunch this series is conditionally convergent, but I get nowhere using the ratio test. What ...
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2answers
35 views

Integration - involving logarithm

I have a slight confusion on how to integrate functions of the form: $$\int\frac{a}{x}dx$$ Suppose we have the following function: $$\int\frac{-2}{x}dx$$ There are two ways we can proceed to ...
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1answer
62 views

Find the product xy.

Given $$ \log (x) + \frac{\log (xy^8)}{(\log x)^2+(\log y)^2} = 2\\ \log (y) + \frac{\log \left(\frac{x^8}{y}\right)}{(\log x)^2+(\log y)^2} = 0 $$ Find the product $xy$ if both $x$ and $y$ are ...
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1answer
27 views

Inverting the exponential decay equation?

How would I algebraically isolate $x$ in the following exponential decay equation, so that $x$ is most easily derived? $$ y = A e^{-\lambda x} \quad $$
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2answers
41 views

Solve simultaneous equations $\log(x-2)+ \log 2=2 \log y$, $\log (x-3y+3)=0$ (Not sure of solutions in book)

Solve simultaneous equations $\log(x-2)+\log2=2\log y$, $\log(x-3y+3)=0$ (Not sure of solutions in book) My method: $\log(x-2)+\log 2-\log y^2=0 \Rightarrow \log\left(\frac{x}{y^2}\right)=0 ...
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2answers
14 views

System of Equations involving Logarithmic Function

Solving a piecewise defined function for real solutions: $x+y=65$ $\log_{2}{x} + \log_{4}{y} = 3$ So far I've changed $\log_{4}{y}$ in terms of base $2$ and then plugged that into the second ...
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1answer
24 views

Solve for $x$ correct to two significant figures, the equation: $4^{2x+1}.5^{x-2}=6^{1-x}$ (Conflicting answer with book)

Solve for $x$ correct to two significant figures, the equation: $4^{2x+1}.5^{x-2}=6^{1-x}$ (Conflicting answer with book) My method: $4^{2x}.4.5^{x}.5^{-2}=6.6^{-x} \Rightarrow ...
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5answers
73 views

Solve for $x$, correct to two significant figures, the equation: $4^{x}-2^{x+1}-3=0$

Solve for $x$, correct to two significant figures, the equation: $$4^{x}-2^{x+1}-3=0$$ My answer: $x\log4=\log3+(x+1)\log2 \Rightarrow 0.602x-0.301x=0.477+0.301 \Rightarrow x = 2.6$ (Conflicting ...
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1answer
54 views

Comparing $3^{1431}$ with $2^{2010}$ without logarithms

I want to compare : $3^{1431}$ and $2^{2010}$ I tried logarithms, $\mathrm{log}_2$ is the way to go. $\mathrm{log}_2(2^{2010})=2010$ $\mathrm{log}_2(3^{1431})=1431\,\log_2(3)=2268.08$ since ...
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0answers
33 views

What happens if you apply the “logarithm of a product” property to a negative number?

We know that $\log_a xy=\log_a x + \log_a y$ and that logarithms of negative numbers are undefined. But what happens if we try to apply this property to let's say $-5$? $\log_a-1*5=\log_a-1+\log_a5$ ...
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1answer
98 views

$\lg_{2} \left( \prod\limits_{a=1}^{2015} \prod\limits_{b=1}^{2015} (1 + e^{\frac{2\pi iab}{2015}}) \right)$? [closed]

How can this problem be solved? $$ \lg_{2} \left( \prod\limits_{a=1}^{2015} \prod\limits_{b=1}^{2015} (1 + e^{\frac{2\pi iab}{2015}}) \right) $$
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1answer
48 views

Prove $\log(n) = O(n)$ using induction

I am using the lecture notes here on page 19 (Algorithm Notes 1) example 1 is the inductive proof of $\log(n) = O(n)$. I understand the base case but I don't understand the rest of the example. ...
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1answer
31 views

Showing an inequality with ln

I have to show that the following inequation is true: $\frac{\ln(x) + \ln(y)}{2} \leq \ln(\frac{x+y}{2})$ I transformed it into $\frac{\ln(x \cdot y)}{2} \leq \ln(x+y) - \ln(2)$ because I thought ...
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1answer
58 views

Is this number in $O(\log(n))$?

Is this number $\big[\log(n) + \sum_{j=1}^{n-1} (\log(j) - (j+1)(\log(j+1)) + j \log(j) +1)\big] \in O(\log(n))$? I simplified it to $\big[\log(n) + \sum_{j=1}^n (-\log(n+1) - j(\log(n)) + 1)\big]$.
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2answers
48 views

$x^y = \exp( \ln(x) \cdot y )$, not a real solution for decimal numbers?

I am trying to understand how to calculate $x^y$ where $y$ is a decimal number, ($2^{2.3}$) According to wikipedia, the 'solution' would be $$ x^y = \exp( \ln(x) \cdot y ).$$ But if we break it ...
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2answers
33 views

Approximate a summation

Approximate $3+ \displaystyle \sum_{x = 2}^{999}\dfrac{3(1000-x)}{1000+x}$. It may help to know that $\ln 2 = 0.69$. I was thinking of doing the integral test to approximate this but I am unsure if ...
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2answers
102 views

showing that the partial sums of $ \log(j) = n\log(n) - n + \text{O}(\log(n))$

I'm trying to show that the partial sums of $\log(j) = n\log(n) - n + \text{O}(\log(n))$ I know that $$\int_1^n\log(x)dx = n\log(n) - n + 1$$ so that this number is pretty close to what I want. Now ...
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1answer
67 views

$\log(xe^{2\pi i})=$?

I am confused about the differences of the properties of natural log in complex analysis and in real region. This question might be a bit stupid, but any answers or explanations of the log properties ...
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2answers
659 views

Roots of ln of a square

Problem: $$ y=\ln((3x-2)^2) $$ State the domain and the coordinates of the point where the curve crosses the x-axis At first sight, you say that the domain is $x>\frac23$ because $\ln$ is ...
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3answers
49 views

Why does $b^{\log_bx} = x$?

Why does $b^{\log_bx} = x$? Can someone break this down by showing me the steps as to why this is true?
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0answers
24 views

Converting log-scaled volume density to number fraction

I have a log scaled volume density distribution, $q_{3,log}$ from which I want to get number fraction, $\Delta Q_0$ with normal scale. So to transform $q_{3,log}$ to $\Delta Q_3$ the used relation is ...
0
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1answer
19 views

Find the dimensions of the rectangle with max area, base on positive x-axis, a side on the y-axis, and a vertex on y = e^(−x^2)

I know that this must maximize the definite integral from 0 to the x value, which would use the derivative of the integral but I'm unsure of how to set up the equation.
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0answers
37 views

logarithmic singularities in contour integration

How to evaluate the contour integral using the residue theorem if there is a logarithmic derivative? For example this: $$\int_C \log\zeta(s)\frac{x^s}{s} ds$$ or even this: $$ \int_C \frac{\log ...
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3answers
55 views

How do I prove that $f(x) = ln(x) − (x − 4)^2$ has exactly two roots.

I assume we take the derivative of the function. I get: $y' = 1/x-2(x-4)$ and I attempt to set it to 0 and solve but get stuck. Any tips?
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4answers
49 views

How do you find $y'$ for $x^y = y^x$?

Using the laws of logarithms: $y\ln(x) = x\ln(y)$, $y = x\frac{\ln(y)}{\ln(x)}$ Is it now quotient rule for the derivative? How is this done?
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4answers
46 views

find $y'$ for $y=(4+x^2)^x$

This differentiation requires the use of natural logarithms (the laws of logarithms), differentiation of logarithms, exponential function differentiation and the power rule. the formula for ...
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1answer
27 views

Solution of $\frac{c - 1}{x - c} + \log \frac{c - x}{x - 1} = 0$ for $x$

I was looking for the maximum of the function $f(x) = \left(x - 1 \right) \log\frac{c - x}{x - 1}$ for $\{x,c\} \in\mathbb{R}^+$, $x\not=1$ (obviously) and $x \le c-1$. The normal way to find such ...
2
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1answer
30 views

When does $\log(\lim_{x\to c} f(x)) = \lim_{x\to c} \log(f(x))$?

When does $$\log(\lim_{x\to c} f(x)) = \lim_{x\to c} \log(f(x))$$ I have seen different things from different sources. For example, this and this. Does $f(x)$ have to be continuous at c, does $\log ...
3
votes
3answers
66 views

Solve the equation,$\sqrt{\log(-x)}=\log{\sqrt{x^2}}$

Solve the equation,$\sqrt{\log(-x)}=\log{\sqrt{x^2}}$(base of log is 10) $\sqrt{\log(-x)}=\log{\sqrt{x^2}}$ $\sqrt{\log(-x)}=\log{|x|}$ Now two cases arise,when $x>0$ and when $x<0$ When ...
2
votes
2answers
30 views

$c$ is the value of $x^3+3x-14$ where $x=\sqrt[3]{7+5\sqrt2}-\frac{1}{\sqrt[3]{7+5\sqrt2}}$.Find the value of $a+b+c$

$a=\sqrt{57+40\sqrt2}-\sqrt{57-40\sqrt2}$ and $b=\sqrt{25^{\frac{1}{\log_85}}+49^{\frac{1}{\log_67}}}$ and $c$ is the value of $x^3+3x-14$ where ...
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1answer
20 views

Find the number of positive integers such that logarithm of whose reciprocals to the base 10 has the characteristic $-2$.

Find the number of positive integers such that logarithm of whose reciprocals to the base 10 has the characteristic $-2$. Let $x$ be a positive integer. Now the characteristic of ...
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0answers
24 views

Solving the curve equation for logarithmic decay using two anchor points.

I would like to have an adaptable logarithmic curve equation that I can then find y for any value of x. I have two points (x1,y1) and (x2,y2). My data requires constant decay (financial discounting ...
0
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3answers
20 views

How do I calculate the following logarithm?

Say I'd like to calculate the following logarithm: $$log_{0,1}{\sqrt {10}\over 100}$$ Using the logarithm rules, I do it this way: $${log_{1\over 10} {\sqrt {10}}} - {log_{1\over 10} {100}}$$ ...