Questions related to real and complex logarithms.

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2answers
51 views

How to solve equation

I have a problem to solve that equation $\log_{4}\left(x\right) = -2x + 9$ i know that answer is 4 but what is step by step solution.
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1answer
32 views

Is O(nlog(2,n)) in O(n^2 )?

Trying to do Big O proofs and I'm stuck on this proof. Need to prove if O(nlog(2,n)) is in O(n^2) After playing around with it I get log(2,n)/n <= c but I'm not too sure what to do after or how ...
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1answer
188 views

Logarithmic inequalities

Full disclosure: This is a homework problem, but my question is regarding a concept that came about during solving the problem, not the actual solution to the problem. Problem: Rewrite as geometric ...
-1
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2answers
31 views

How to solve the logarithm equation? [closed]

$\log_{2}(\log_{x}25)=1$ How should I solve it? No idea!
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1answer
61 views

CAGR, log of negative numbers

I am trying to calculate the Compound Annual Growth Rate (CAGR) of a number of companies using the Geometric mean of the annual growth of their EPS. Some of the EPS values are negative(loss making) ...
3
votes
2answers
42 views

Solve an equation involving logarithms

$ (\log_{10}x)^2 = 3 \log_{10}x$ Should I do it in this way: $\log_{10}x^2=3\log_{10}x$ $\frac{\log_{10}x^2}{\log_{10}x}=3$ Is it right? If I continue like this, I get only one answer whereas my ...
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1answer
58 views

Need help to solve this equation: log cosx(4) * log cos^2 x(2) =1

Equation: log cosx(4) * log cos^2 x(2) =1 *cosx, cos^2 x - base 4,2 -numbers cosx>0 and not 1 then cosx is in (0;1) i've tried: if cosx=t then log t(4) * log t^2 (2) =1 ; 2log t(2) * ...
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2answers
59 views

Comparing two decibel values

For my son's science fair project, we are measuring wi-fi signal strength in decibels, a logarithmic scale. We want to determine the relative strength of two values. I think that a value of -60 is ...
1
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1answer
65 views

Creating a function with logarithmic growth

I have some knobs with an internal value of $0$ to $1$. These represent a value in a range, like $1$ to $1000$. Case in point, I would like to be able to change the scale/growth of the display value. ...
0
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1answer
689 views

Can I break up $\log(a - b)$?

For constants $a$ and $b$, I know that I can break up $\log(a/b)$ into $\log(a) - \log(b)$. Can I conveniently break up $\log(a - b)$ somehow into several terms?
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1answer
47 views

Normalized Mutual Information results in log(0) with non-overlapping clusters - how to deal with that?

I want to evaluate how well my flat soft clustering method works, compared to a gold standard. After some research I found that Normalized Mutual Information would most likely be a good measure, for ...
0
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1answer
88 views

Undoing the Natural log after integrating $ln \frac{ \sec{x} \tan{x}}{3x+5}dx $

Since beating my head against a brick wall is so fun, I kept working on this old integral $\int \frac{ \sec{x} \tan{x}}{3x+5}dx$ . I think I have finally found a way to do it. Here goes. $$ \int ...
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2answers
17 views

Calculating pOH

If [OH-]=10^-pOH and [OH-]= 0.003 then what does pOH equal? I know this is simple but I just can't figure out how to do this calculation. Any help would be appreciated.
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2answers
64 views

Are the sums equal to each other?

They are $2$ different results for the integral $$\int xe^{2x}\sin\left(\frac x3\right)\,dx$$ $\displaystyle\frac{-3}{1369}e^{2x}\left(3(35-74x)\sin\left(\frac x3\right)+(37x-36)\cos\left(\frac ...
2
votes
2answers
85 views

Evaluating limit using logarithms.

Evaluate the following limit. $$ \lim_{x\to \infty} (\ln\ x)^{\frac{1}{x}} $$ What i have tried: $$ \ln\left[\lim_{x\to \infty} (\ln\ x)^{\frac{1}{x}}\right] $$ $$ \lim_{x\to \infty} \ln(\ln ...
0
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0answers
12 views

2 line segments of similar lengths cut off by logarithmic functions

We have the function $f(x) = \log_{1/3}(x+3), g(x) = 2 - \log_{1/3}(x), h(x) = -3+\log_{1/3}(x-1)$. The line $y=p$ with $-1 < p < 0$ is divided into 2 line segments with equal length by ...
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3answers
62 views

Estimating the natural logarithm from both sides: $1/(a+1)<\ln((1+a)/a) <1/a$

I must prove that for all $a>0$ $$\frac{1}{a+1}<\ln{\frac{1+a}{a}}<\frac{1}{a}$$ Can anyone help me?
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0answers
52 views

When the system of equations below had a solution?

The system of equations is $$\begin{cases} \frac{c_1}{1-x_1}+\frac{c_2}{1-x_2}+\frac{c_3}{1-x_3}=0\\ \frac{c_1}{k-x_1}+\frac{c_2}{k-x_2}+\frac{c_3}{k-x_3}=0\\ ...
1
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1answer
119 views

I wonder whether the system of equations and inequations below have a solution.

I wonder whether the system of equations and inequations below have a solution. If there are solutions, what are they? A numerical solution is also desired. $$\begin{cases} ...
0
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3answers
63 views

Logarithm properties doubt

The problem is $\log (5.64)^4$. According to the properties and laws of exponents, $\log (m^r) = r \log (m)$. But since the exponent is outside of the parenthesis in this problem, does it solves by ...
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votes
2answers
111 views

How to take the Limits of Logs

How would you take the limit of $$\frac{\log(n!)}{\log(n^n)}$$ as $n\rightarrow\infty$. I believe you have to remove the log raising it to their base. Is this correct ? Thanks.
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1answer
58 views

Is $n^{\log c} = c^{\log n}$ true?

Is $n^{\log c}$ the same as $c^{\log n}$? If so, please explain.
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2answers
43 views

Absolute values in logarithms in a solution of differential equation

How have the moduli signs disappeared in the following step: $$\frac1{k}\left(\ln|g+kv| - \ln|g+ku|\right) = -t$$ Therefore $$ \ln\left(\frac{g+kv}{g+ku}\right) = -kt$$ $g$, $k$ and $u$ are ...
0
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1answer
187 views

Proving functions to be Big Oh

How do I determine if there exists a function $f$, such that \begin{equation} f(n) = {\mathcal O}(\log n), \end{equation} but \begin{equation} 2^{f(n)} ≠ {\mathcal O}(n). \end{equation} Is ...
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2answers
29 views

How do I calculate this logarithmic expression?

What I'm not sure about is the power of two above the logarithm. I just wanted to verify I'm calculating correctly Do I do these steps...? 1 - Take absolute value of variable AL 2 - Take log base ...
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1answer
28 views

Is there a seperate something in front logarithm that is raising a base to a power?

I am trying to solve a problem with the following form $$e^{\displaystyle A\log(x)}$$ $e^{\log(x)}$ is simply $x$, but how do I go about separating the $A$?
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votes
3answers
89 views

What's a good class of functions for bounding/comparing ratios of complicated logarithms?

I have this goofy series $\sum \limits_{n=2}^\infty \frac{ \log_2 \left[ n \log_2^2 n \right]}{n \log_2^2 n}$ that Wolfram Alpha tells me diverges by the comparison test (and indeed, in the larger ...
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1answer
158 views

How to solve $\log \sqrt[3]{x} = \sqrt{\log x} ?$

How to solve $$\log \sqrt[3]{x} = \sqrt{\log x} $$
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1answer
44 views

How do I simplify and calculate this inequality?

$\log(x^3) > |x-1|$ I can't figure out how to go about solving this inequality, besides this one step: $3\log(x) > |x-1|$
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2answers
63 views

Solve the Logarithmic Equations for x, please.

This one is an exponential equation that I can't figure out.. $7^{x-2} = 5^{3-x}$ These two are logarithmic equations that I'm also having trouble with.. $\ln \sqrt[3]{x-6} = -2$ ...
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1answer
55 views

Prove if $f(x) = \ln\left(1-\frac{1}{x^2}\right)$ then $f(2)+f(3)+f(4)=\ln(5/8)$

I have that: $$f(x) = \ln\left(1-\frac{1}{x^2}\right)$$ I need to prove that $f(2)+f(3)+f(4)=\ln\left(\frac58\right)$ Indeed, I proved that $f(2)+f(3)+f(4)=\ln(3/4)+\ln(8/9)+\ln(15/16)$ But ...
3
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3answers
55 views

How do we prove that $a^{\log{b}} = b^{\log{a}}$ for $a > 1$ and $b > 1$?

I have tried using the change of base formula, but can't quite complete the equality: $$ a^{\log{b}} \\ a^{\frac{\log_a{b}}{\log_a{a}}} $$ How do I get the base of the exponent to be b?
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1answer
58 views

Growth of |logx| versus of 1/x

Do you think there is a number k s.t. $\int_{(0,\infty)} \frac{|log(x)|^{k}}{x}d\mu$ will converge,where $\mu$ is the Lebesgue measure? If you don't know ,can you at least give me some reference for ...
2
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3answers
90 views

$4\log_2(n)=n$ How to Solve for $n$?

As the title suggests, my log skills are pretty lacking. Need to learn how to get from $4\log_2(n)=n$ to $n=16$ ($\log$ base $2$). I've searched Google and it seems I am missing some core concept ...
2
votes
4answers
115 views

How can i calculate this limit

the limit is $$\lim_{x \to 0} {1 \over \ln(x+1)} - {1 \over x}$$ The problem is i don't know if i can calculate it normally like with a change of variables or not . Keep in mind that i'm not ...
1
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1answer
47 views

If$ a+b-1=1+\frac{ln(2^a-1)}{ln4}+\frac{ln(2^b-1)}{ln4}$ then $a=b$?

If $$a+b-1=1+\frac{ln(2^a-1)}{ln4}+\frac{ln(2^b-1)}{ln4}$$ where $a,b>0$ are real numbers and ln is $log_e$, then is a=b?
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1answer
21 views

Using log of function to determine orders of growth

If I have functions $f(n)$ and $g(n)$ and I would like to determine $f(n) \in \Omega g(n)$ and/or $f(n) \in \Theta(g(n)$. Does proving $\log(f(n)) \in \Omega \log(g(n))$ imply $f(n) \in \Omega g(n)$? ...
2
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2answers
128 views

How do you solve $x^{\log x}=100x$

How do you solve $x^{\log x}=100x$? Can you please thoroughly explain the left side of the equation. Please explain very clearly because I have only been learning logarithms for about a week.
2
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1answer
71 views

Find the Solution of the Exponential Equation?

How do I solve $5^x = 4^x+1$? I understand how to solve for $x$ when there is one exponent, but I don't know how to solve when there is an exponent on both sides of the equation..
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5answers
140 views

Proving $\log x^2 = 2\log x$

How does $\log x^2 = 2\log x$? Can you do a proof please. I know that this is true but I don't know why.
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2answers
67 views

Find $x$ in $\log x^2 = (\log x)^2$

Find $x$ in $\log x^2 = (\log x)^2$. I couldn't find x.
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2answers
59 views

Find the solution to this equation

The equation is the following $2^x-x^2-100=0$ It was handed to me today by a friend student of mine. I hope you enjoy it .
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2answers
69 views

Can I add $\log$ to both sides of inequality such way?

Can I add $\log$ to both sides of the following inequality $$f(n) \leq cn^k$$ and get $$\log (f(n)) \leq kc\log n$$ I know that by rules the result inequality should be like this $$\log(f(n)) \leq ...
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4answers
143 views

Help me to Prove that log2 3 is irrational. [closed]

seemingly simple homework assignment, help? Was never the best with logarithms, how would I go about proving? Sorry the question read IRrational!
2
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1answer
46 views

Simplifying an expression using a logarithm

I have the following expression $$\frac{1}{1+\rho}(1+n)^{(1-\sigma)}*(1+\gamma_{A})^{1-\sigma}<1$$ and have to use logarithms to get the following $$(1-\sigma)(n+\gamma_{A})<\rho$$ Could ...
0
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2answers
51 views

logarithm and exponent computation performance

Using glibc on a x86 processor, which takes more CPU time? $a\ log\ b$ or $b^a$? For which values of $a$ is one faster than the other? Optional: Does the base used matter? See also: What algorithm ...
5
votes
2answers
248 views

Proof $e^x = \exp(x)$?

Define $$\ln (x) = \int^{x}_{1}\frac{1}{t}$$ Assume I have proven that $\ln x$ is one-to-one and therefore has an inverse $\exp (x)$. Define $e$ as: $\ln e = 1$ Now, if you have no other notion ...
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vote
3answers
62 views

How can you prove this equality?

I am trying to figure out the following equality, but cannot seem to get anywhere. I tried integrating by parts, but that blew up when I set u = (log x)^n and tried to take log (0). I also tried ...
11
votes
5answers
2k views

Why is it trivial that $\left(1+\frac{2\ln3}{3}\right)^{-3/2}\leq\frac{2}{3}$?

Can someone tell me why $$\left(1+\dfrac{2\ln3}{3}\right)^{-3/2}\leq\dfrac{2}{3}$$ is trivial because for me its not and I will need to do the calculation to see it.
2
votes
4answers
124 views

Showing $\frac{x}{1+x}<\log(1+x)<x$ for all $x>0$ using the mean value theorem

I want to show that $$\frac{x}{1+x}<\log(1+x)<x$$ for all $x>0$ using the mean value theorem. I tried to prove the two inequalities separately. $$\frac{x}{1+x}<\log(1+x) \Leftrightarrow ...