Questions related to real and complex logarithms.

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0
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1answer
251 views

Interesting problem in logarithms

I know this place isn't for math problems/homework, and believe me I've been trying for a long time to solve this problem (45 mins to 1 hour) and besides I think many would find this useful or at ...
0
votes
2answers
41 views

I'm not great with logarithms so I'd appreciate some help with the following

How is it that $n^{\frac{1}{\log n}} = 10$. I understand that $10^{\log a} = a$ but I don't know how to make the correct algebraic manipulations. Note: Assume $log$ is base 10
1
vote
1answer
157 views

Forward Algorithm Hidden Markov Model matrix help [Discrete]!

So this may seem like a bioinformatics question but it is the math part that is giving me trouble. I'm using a Python package called YAHMM to model DNA sequences. I created a model with two states (...
0
votes
2answers
34 views

Is a dollar gain truly equal to a dollar lost?

Under the context of the asymmetrical nature of gain to a loss, as shown below, is 1 dollar gained truly equal to 1 dollar lost? If not how would you go about calculating the equalization ratio at ...
2
votes
3answers
47 views

Integral of $\frac{1}{2x}$

The integral of $\frac{1}{2x}$ is $\frac{\ln(x)}{2}$, but can't it also be $\frac{\ln(2x)}{2}$ or $\frac{\ln(3x)}{2}$? Is there a special reason for $\ln(Ax)$ to have identical derivatives?
8
votes
6answers
208 views

Why is $\ln(x^x)=x\ln(x)$ valid?

I know that $\ln(x^k)=k\ln(x)$ for any constant $k$, but why is $\ln(x^x)=x\ln(x)$. The exponent $x$ is not constant.
2
votes
4answers
86 views

How to solve $4x-\log(x) = 0$

I have a problem solving this equation: $4x-\log(x) = 0$. I can't seem to get this equation to a simpler form featuring $\log$s only or getting rid of the $\log$. Is there a way to solve it without ...
0
votes
1answer
34 views

Best fit in logarithmic chart

I have several variances ($\sigma^2$) which value depends on the velocity ($v$). As you can see in the graph, if increase the velocity, the variance does the same. I am studying this dependency, but ...
2
votes
2answers
30 views

Evaluating Logarithmic Expressions

Evaluate: $$\log_4 \left(\dfrac{1}{256}\right)$$ I am not sure how to approach this since there is nothing set equal to it.
3
votes
1answer
253 views

Integration of (Tsiolkovsky) rocket equation

The (Tsiolkovsky) rocket equation states that the velocity of a rocket can be calculated as $$v(t) = v_0 \ln\frac{m_0}{m_0 - \dot m t}$$ where $m_0$ is the starting mass, $\dot m$ is the (constant) ...
1
vote
1answer
34 views

How to normalize data in another scale?

Let $A$ be a set of values $\{a_1,a_2,a_3,a_4,a_5\}$ where $a_1 = 2$, $a_2 = 1$, $a_3 = 4$, $a_4 = 1$ and $a_5 = 2$, so, the $avg(A) = 2$. I'm looking for a normalization where the values below the ...
2
votes
0answers
117 views

Finding how many solutions does $f(x)=\ln x-kx$ has for $k>\frac 1 e$ and logarithmic inequality question

Find how many solutions does $f(x)=\ln x-kx$ has for $k>\frac 1 e$. $f<0$ at $x\to \infty$ and $x\to 0$. The derivative has a solution only at $x=\frac 1 k$. So place that point in $f$ and we'...
0
votes
3answers
63 views

Simplifying Quadratic Equations In Logarithmic Form

$log_{10}(x^2-x-7)=0.1$ $log_{10}(x-8)=1-log_{10}(x+1)$ $log_{10}(x+9)=1+log_{10}(x+1)-log_{10}(x-2)$ Note: I solved them as follows: $x = 3, -2$ but the textbook i'm using said there was no ...
3
votes
1answer
31 views

Prove convergence of this generalized integral

Prove the convergence of $$\int_0^1 \left[\ln\left(1+\frac1x\right)\right]^a\mathrm dx$$ for $ a>0$.
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votes
2answers
100 views

Compute $\lim\limits_{n\to \infty} \ln(3n+7) - \ln(n)$

The reason why I'm having trouble with this problem is because it involves natural log (ln) and I need to find the limit. I need to find $\lim_{n\to\infty} \ln(3n+7)-\ln(n)$. I noticed that as $n$ ...
1
vote
2answers
62 views

Solve the logarithmic equation by $x$

Solve the eqation for all real $x$: $\log_2(x^2+7)+\log_3(x+6)=6$. What I tried: $\log_2(x^2+7)=a$ and $\log_3(x+6)=b$, then $a+b=6$ and $2^a=3^{2b}-4\cdot3^{b+1}+43$. But the problem is $a$ and $b$ ...
0
votes
2answers
550 views

Solving for n, n is an exponent.

If you have a sequence of random numbers ranging between 1 and 64, what is the length of a sequence that will give a 98% chance of having at least one ( 1, 2, or 3) in the sequence? Here is the ...
1
vote
1answer
21 views

Multi-variable calculus involving $\ln$

I am having difficulty with differentiating this equation with respect to $y$: $$ W= x^{y \ln(z)}. $$ Differentiating calculators are giving me the answer $$\ln(x) \ln(z).x^{y \ln(z)}$$ But I can'...
0
votes
1answer
42 views

Is it true that $x^n <\epsilon \Rightarrow n < \frac{\ln \epsilon}{\ln x}$?

Let: $0 \lt x \lt 1$ $\epsilon > 0$ I need to show that there exists an $N(\epsilon,x)$ such that: $n\ge N(\epsilon,x) \Rightarrow x^n < \epsilon$ This is what I've tried: $x^n <\...
1
vote
1answer
30 views

How to “see” that this expression is $>0$.

$N \in \mathbb N$. $\displaystyle\int_{N-1}^N \left(\dfrac{1}{x} - \dfrac{1}{N}\right) dx>0$ This is the finish of a proof, a modification of $\log N-\log (N-1) -\frac{1}{N}$. Calculating it ...
4
votes
2answers
53 views

If $\log_{12}54=a$ then $ \log_{6}12=?$

I am given $$\log_{12}54=a$$ So what will be value of $ \log_{6}12?$ I used base changing theorem and wrote expression as $$\frac{\log_{6}54}{ \log_{6}12} =a$$ And then $$ \frac{1+\log_{6}9}{ a} = \...
0
votes
1answer
45 views

How to solve equations containing logarithms and exponentials

Equation 1: $x+e=e^x$ According to Wolfram alpha : Solution of x $\approx$ -2.6 and 1.4 Equation 2: $x-e = \ln(x)$ According to wolfram alpha, Solution for x $\approx$ 0.07 and 4.1 How does ...
0
votes
1answer
39 views

Properties of Geometric Series

If we have a geometric series $(x_1, x_2, ..., x_{n-1}, x_{n})$ of reason $q$, we can determine the general term formula to be: $x_{1}q^{n-1} = x_{n}$ But by taking the logarithm of the equation we ...
0
votes
1answer
13 views

Linking summations with their correct function(s)

Guys can you please guide me step by step on how to link given functions with the functions to choose from. So for example a function $g(n)\in \Theta n^2$ and if there is no match then you say there ...
1
vote
4answers
64 views

If $a^x=b$, then $ x=$?

Stupid question, I know, but I couldn't remember nor find information by googling on how to find the exponent of $a$ that gives $b$ as the result. If $a^x=b$, then $x=log_a b$ but how do you find $x$?
2
votes
5answers
290 views

How to determine the monthly interest rate from an annual interest rate

I have a calculation which gives me the annual interest rate if I already know the monthly interest rate as follows: (Monthly interest rate + 1)^12 In this case I ...
2
votes
2answers
116 views

Solving inequality involving square root and division by logarithm: $\sqrt n<\frac{n}{\log(n)}-2$

I would like to solve the inequality $\sqrt n<\frac{n}{\log(n)}-2$. for some reason I had never done this before. This is clearly the same as $\frac{n}{\log(n)}-\frac{n}{\sqrt{n}}>2$. Which is ...
1
vote
0answers
40 views

Homework : Anti log expression

I have this expression $x(r) = y(a)r^a$ where $r$ is a random variable and I want to express the expression in terms of $r$. The objective is to substitute the variable $r$ into the pdf of $r$, $p(r)$....
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1answer
60 views

Solving natural logarithms with absolute value

Question from my text: $e^{4x-2014} - 7 = |-3|$. I've never seen this before and my text is useless! Thank you!
1
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0answers
22 views

Notation question

What exactly do you think $\ln^rn$ means in this context? "Prove that the relation $\tau(n) = O(\ln^rn)$ is false for all fixed powers $r$." Where $\tau$ is the divisor function.
0
votes
1answer
33 views

Log inequality- is $(\lceil\log x\rceil - \lfloor\log m\rfloor)\cdot m+2^{\lfloor\log m\rfloor+1}\leq m\cdot(\lceil\log\frac{x}{m}\rceil+2)$?

I'm having some hard times making a tight analysis of the memory requirements for my algorithm. I want to show the following inequality, which will show my data structure can use about 2 bits per ...
2
votes
2answers
64 views

Log inequality - is $\lceil\log x\rceil - \lfloor\log y\rfloor\leq \lceil\log\frac{x}{y}\rceil+1$

Is it true that $$\forall x>y\in\mathbb N:\lceil\log_2 x\rceil - \left\lfloor\log_2 y\right\rfloor\leq \left\lceil\log_2\frac{x}{y}\right\rceil+1$$? I reached this inequality when further ...
1
vote
2answers
206 views

properties of logarithms ln12-ln2=ln6

I checked wolframalpha and it says that ln12-ln2=ln6. How? i tried to do: ln12=ln(2*2*3) which may be 2ln(2*3) (which is probably wrong). I need help. EDIT: Ok, thanks. Actually i could have just ...
0
votes
1answer
25 views

Inverse a simple equation

Consider equation $y = x\cdot 2^x$ Can you write $x$ based on $y$ ? Is it possible ? Thanks
0
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4answers
40 views

using logarithms to solve the following equation to find x

$9^{2x} = 27^{1-x}$ ?? I'm really struggling with this questions. I appreciate your help and if you can please show me your working out so I can understand it too,
1
vote
2answers
237 views

formula for logarithmic spiral on a linear level

I am trying to plot the contents of a circle, which include geometric elements and spirals, on a linear graph. For example, take a circle, take the beginning and the end and make it straight. What ...
0
votes
1answer
64 views

Summation of harmonic series. [closed]

I'm trying to figure out how to answer this linear algebra question and can't figure it out. Can someone please explain it to me? Thanks a bunch! Here's the questions:
4
votes
4answers
113 views

Weird integration issue: $\ln(x+1)=\ln(2x+2)$ ?!

Weird integration issue: Using $(\ln[f(x)])'=\frac {f'(x)}{f(x)}$ we get that $\int \frac{2\,dx}{2x+2}=\ln(2x+2)$. Yet, $\int \frac{2\,dx}{2x+2}= \int\frac{dx}{x+1}=\ln(x+1)$ using the same rule as ...
2
votes
2answers
137 views

Does there exist any positive integer $n$ such that $e^n$ is an integer (to show $\log 2$ is irrational)?

Does there exist any positive integer $n$ such that $e^n$ is an integer ? I was in particular trying to prove $\log 2$ is irrational; now if it is rational, then there are relatively prime integers ...
2
votes
1answer
38 views

Deriving logarithm in exponent

Im attempting to take the derivative of $n^{log_2(n)}$, but the answer I'm getting is different from http://www.derivative-calculator.net/.. this isnt highschool math homework, I'm trying to use L'...
2
votes
1answer
84 views

Proving a series diverges

Hello I am trying to prove the following series diverges $$\sum_{k=1}^{\infty} \ln\left(1+\frac{(-1)^{k+1}}{\sqrt{k+1}}\right)$$ This series alternates around 0 and goes to zero but fails the ...
3
votes
3answers
73 views

What $n^{\frac{1}{\log_2n}}$ means?

I was confused with about the $n^{\frac{1}{\log_2n}}$ expression. I am not sure how to make mathematical sense of it - i.e. express it in another way for easier understanding. I tried to plug in some ...
1
vote
1answer
115 views

$a$ has a square root modulo $p$ if and only if its discrete logarithm log$_{g}(a)$ modulo $p - 1$ is even

Questions: Let $p$ be an odd prime and let $g$ be a primitive root modulo $p$. Prove that $a$ has a square root modulo $p$ if and only if its discrete logarithm log$_{g}(a)$ modulo $p - 1$ is even. ...
2
votes
2answers
313 views

Solving a second-degree exponential equation with logarithms

The following equation is given: $8^{2x} + 8^{x} - 20 = 0$ The objective is to solve for $x$ in terms of the natural logarithm $ln$. I approach as follows: $\log_8{(8^{2x})} = \log_8{(-8^{x} + 20)}...
1
vote
1answer
56 views

How to simplify logs and powers?

Is there any way to simplify $(\log a)^{\log b} = c$? And even this $(\log x)^y = z$? And also this $(\log m)(\log n) = p$ (which is essentially $\log m^{\log n} = p$) I was trying to simplify some ...
5
votes
4answers
165 views

Can someone explain why $x^{\log(a)} = a^{\log(x)}$?

I'm trying to see why the below is true. $$ x^{\log(a)} = a^{\log(x)} $$ Anyone here know why this is? Thank you.
0
votes
4answers
109 views

Does $\ln|x+2|=\ln|2x+4|$ and if so why so? [closed]

Is $\ln|x+2|=\ln|2x+4|$? Is this right? I saw something earlier saying this was correct; my first instinct was no.
2
votes
1answer
30 views

how to calculate this logarithmic function?

Im having trouble in graphing this log function: $y=\log _{1/4}\left|x^2-5x+6\right|$ I found the intervals: $(-\infty, 2)$, $(2,3)$, $(3,\infty)$ Should I just give $x$ values and find $y$ to graph ...
1
vote
2answers
175 views

Find the general expression from the antiderivative

I am having trouble computing the original function. Question states: Let $f$ be a differentiable, positive function, such that $$f'(x)=x*f(x)$$ for all real numbers x. A) Find the general ...
0
votes
1answer
261 views

Confusion: dB to scalar and from scalar to dB

Assume we have $$N_1=5 \text{ dB}$$ $$N_2= - 110 \text{ dB}$$ Then we have $$Y=N_1+N_2=-105 \text{ dB} $$ If I convert to scalar then $$10 \log(X) = -105 \rightarrow X=10^{-10.5}$$ Let me start the ...