Questions related to real and complex logarithms.

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1answer
49 views

Log-linearizing $Y_t=\int_0^1 F(X_{it}) di$

I want to prove that log-linearizing the expression $Y_t=\int_0^1 F(X_{it}) di$ yields: $$Yy_t \approx F'(X)X\int_0^1 x_{it} di$$ Where: $\{X_{it}\}_{i \in (0,1)}$ is a continuum of strictly ...
4
votes
4answers
90 views

Solving logarithmic equation $2\log(x) + 1 =\log(19x+2)$

I'm stuck trying to solve $$2\log(x) + 1 =\log(19x+2)$$ I know the solution has to be $x = 2$. However I can't find the manual steps (Wolfram doesn't know the manual steps either). This is all I ...
1
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2answers
30 views

Simplifying using the rule of logs

How would you simplify: $\left[\frac{16}{5}\ln(x + 2) - \frac{1}{5}\ln(x - 3) - \ln x \right]_4^6$ and put it in the form $\ln\frac{m}{n}$. Stating the values of m and n. Note: $a\ln b = \ln b^a$ ...
0
votes
2answers
20 views

Exponential/Logarithmic system of equations

I'm given the following equations of which I need to find the solutions for algebraically. $ab = 8$ $2^a=c$ $c^b = 256$ My first thought was to use logarithms, but I got a bit lost in doing so ...
2
votes
2answers
79 views

I am not sure, but it's an equation…

$$2x + 2^x = 4$$ So it's clear that $x$ will equal $1$ but how can it be solved through an algebraic method to be able to determine $x$ with more complex numbers. I've tried to transform the exponent ...
1
vote
0answers
31 views

Fractional logarithm operator

Does there exist a fractional logarithm operator? Something like this: $$L_0(x) = x$$ $$L_1(x) = log(x)$$ $$L_{0.5}(x) = ???$$ This is the motivating situation: consider perception of sound, which ...
0
votes
1answer
25 views

Finding log base when given only a graph

I have the function f(x)=log_bx is shown as a graph and the only two points are (5,1) and it's asking me to find the base of the logarithmic function.
2
votes
5answers
108 views

Solve equations like $3^x+4^x=7^x$

How can I solve something like this? $$3^x+4^x=7^x$$ I know that $x=1$, but I don't know how to find it. Thank you!
10
votes
1answer
97 views

Is it true/known/important that $(\log p_n)/n$ is nonincreasing, where $p_n$ is the $n$th odd prime number?

First thing first, I would like to apologize in advance for my poor knowledge of Maths and English. I'm an Italian student and after asking to all the mathematicians and Maths teachers in my town, I ...
1
vote
1answer
44 views

How to solve logarithm word problem given the exponential equation?

The question is The world population in 2000 was approximately 6.08 billion. The annual rate of increase was about 1.26%. The function that models this is $$ y = 6.08(10)^{.0052t}$$ where y ...
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votes
2answers
23 views

How do I solve an exponential equation using Natural Log.

I understand how to solve a problem such as $6^{2x+3}=11$ by using natural log, but the question thats tripping me up is $3^{1-x}=7^x$. Mathway and Wolfram Alpha tells me what the answer is, but I ...
0
votes
1answer
63 views

Is this true (Natural Logarithm)?

Is it true if I say, when we know that $\text{D},\beta\in\mathbb{R}$: $$\text{D}=e^{-\pi\frac{\beta}{\sqrt{1-\beta^2}}}\Longleftrightarrow ...
0
votes
1answer
42 views

Difference between gaussian and log-normal distribution

I have a random variable say X that is a Gaussian distributed with mean equal to zero dB. When I convert it into linear domain, i.e from dB to linear, does it imply that the resulting variable is ...
0
votes
1answer
101 views

Limit evaluate $\lim_{x\to0}{{\frac{\ln(\cos(-5x))}{\ln(\cos(-3x))}}}$ [duplicate]

Please help me with this limit without using L'Hôpital's rule. I would by happy if you use simple solving. Thank you as much as I can ;). $$\lim_{x\to0}{{\frac{\ln(\cos(-5x))}{\ln(\cos(-3x))}}}$$
0
votes
0answers
44 views

x raised to the power of x raised to the power of logarithm base 3 x

Have a question that has me a little stumped. I know that this equation:$$x^\left(log_3x + x^\left(log_3x\right)\right) \neq 162$$ Simplifies to: $$x^\left(log_3x\right) \neq 81$$ My question is how ...
1
vote
5answers
59 views

Derivative of $1-5^{-x}$

What is the derivative of $y=1-5^{-x}$. Any help is greatly appreciated! I have tried using logs, but I don't think it is correct; $$y=1-5^{-x}$$ $$\ln(y)=\ln(1) +x\ln(5)$$ $$y =x\ln(5)$$ and hence ...
0
votes
1answer
37 views

How do I solve $3^{\ln{2}} \times x^{\ln x + \ln 6 + 1} = \frac{3e^2}{4}$

I've been going at this question for 2 hours, my teacher wants us to solve for x without a graphing calculator. \begin{equation} 3^{\ln{2}} \times x^{\ln x + \ln 6 + 1} = \frac{3e^2}{4} ...
0
votes
1answer
36 views

How to graph log equation if given the following?

So the question is $$ Y= log(-x+2)$$ So I factored as my teach told the class that $x$ can only be subtracted so I factored out $$ Y = log(-(x-2))$$ Information I got from the following equation ...
3
votes
3answers
138 views

What is the inverse function of $e^x +x$?

As the natural $\log(x)$ function is the inverse of the exponential $e^x$ and $\log(x +1)$ is the inverse of $e^x - 1$, what it the inverse of $e^x + x$?
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2answers
69 views

Is there a difference between $N\log{\log N}$ and $N\log^2N$

I am trying to compare the growth rates of functions to review my understanding of basic Algorithms. The text asks to compare: $$N\log \log N$$ and $$N\log^2{N}$$ Are they not the same function?
0
votes
1answer
52 views

How to simplify $\ln{\left(x + \ln{\left(x + \ln{\left(x + …\right)}\right)}\right)}$.

I have tried the following: $s = \ln{\left(x + \ln{\left(x + \ln{\left(x + ...\right)}\right)}\right)}$ $s = \ln{\left(s + x\right)}$ $e^{s} = s + x$ However, I am unsure as to how to proceed. ...
0
votes
1answer
57 views

How to find the solution to this summation

This was a question asked in our exam and we have to write a code for it. We have to find the summation of following series $\log(\sum_1^n (e^{x_i}))$ where $1 < n < 10^6$ and $0 < x_i < ...
0
votes
1answer
40 views

How to solve common log problem given the following with the help of calculator?

the question is $$\log_{10}(x + 8) + 6 = 8$$ $$\log_{10}(x+8) = 2$$ $$10^2 = x + 8 $$ I just turned it into exponential form. Now my question is if i did this right at all? and if i did how would ...
1
vote
1answer
23 views

Can O(n + logn) be called O(n)?

I know that O(n + 5) would be classified as a worst-case runtime of O(n), but can I do that with logn too?
1
vote
1answer
41 views

How many digits will $ab^c$ have?

How many digits will $ab^c$ have? I know that the digits of $b^c$ is calculated so: $$\lfloor c \log_{10}b \rfloor +1$$ but what about $ab^c$ ?
0
votes
3answers
23 views

Exponentials with three variables: solving for an equation

$2^{x}=5^{y}=100^{z}$ Find $z$ in terms of $y$ and $x$. The term $z$ should be a function of $x$ and $y$, i.e.: $z(x,y)$. All I could get were recursive attempts.
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3answers
65 views

Logarithm of imaginary numbers?

How do I solve this question? I tried using the quadratic formula on the question equation and got $x_1 = 0.25 +1.089724..i = \ln r$ $x_2 = 0.25 -1.089724..i = \ln s$ I know $\ln x = \log_ex$, ...
1
vote
2answers
94 views

Express this logarithm in terms of a and b

How do I express $\log_52$ in terms of $a$ and $b$ if: $\log_62 =a$ and $\log_53 =b$ I've tried: Converting the $a$ and $b$ equations to fractions, and substituting $\log2$ and $\log5$ with ...
0
votes
3answers
95 views

Solve simultaneous logarithmic equations with different bases?

How do I solve these simultaneous equations? $2 log_x y+2log_yx = 5$ $xy=8$ I've tried to convert the first formula to fraction form and continue from there, but I can't seem to get anywhere. I've ...
0
votes
0answers
64 views

relation between parameters so that sequence is fundamental

Need help with this... Let $$f,g : \mathbb{N^*} \to \mathbb{R_+^*}$$ be functions respectively defined by $$ f(n) = \ln(n+1) - \ln n$$ and $$ g(n) = \sum\limits_{k=n}^{2n-1}f^3(k), \forall n \in ...
1
vote
3answers
47 views

Prove that: $\lim\limits_{n\rightarrow \infty}({\frac{10^{\log_2{\log_2{n}}}}{\log_2{n}}})=\infty$

Prove that: $$\lim\limits_{n\rightarrow \infty}\left({\frac{10^{\log_2{\log_2{n}}}}{\log_2{n}}}\right)=\infty$$ I've tried applying L'hopital to no avail.
7
votes
2answers
109 views

Solving a logarithmic equation $\log_2 (2^x-1)+x=\log_4 (144)$

I need to solve this: $$\log_2 (2^x-1)+x=\log_4 (144)$$ I can work out that $x=\log_2 (2^x)$ and $\log_4 (144)=log_2(12)$ but I'm stuck after that.
2
votes
2answers
151 views

How to find the indefinite integral for a natural log being divided by x?

I've done many examples in the math book but none of them have a natural log as a numerator. Here's the question. $$\int\frac{(\ln\ x)^7}{x} dx$$ I am given these 2 properties, where $u$ is a ...
1
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3answers
48 views

Logarithmic Equation: How to solve for x

Equation: $$\log_a (x) + \log_a (x-4) = \log_a (x+6)$$ Progress $$\log_a (x^2-4x) = \log_a (x+6)$$ $$x^2-5x-6=0$$ Delta $$x1= 6$$ $$x2=-1$$
0
votes
1answer
36 views

Limit logarithm $\lim_{n \to \infty}(4n-6)[\ln(2n+5)-\ln(2n-7)]$.

please any advice for this limit? $$\lim_{n \to \infty}(4n-6)[\ln(2n+5)-\ln(2n-7)]$$ Thanks for any advice.
1
vote
1answer
30 views

Can one realize the real part of every entire function $f$ as $\ln| g|$ with $g$ entire?

Let $\Re$ denote real part and $|\cdot|$ absolute value. Does there exist, for every entire $f$, an entire $g$ such that $\Re f = \ln |g|$ ?
0
votes
1answer
111 views

Radioactive Decay formula is $A=A_0e^{-kt}$. How many years until 10 grams decay so that only 8 remain

I have been trying this question for hours and come to a dead end every time... Consider the radioactive decay formula $A=A_0e^{-kt}$ where $A$ is the amount of radium remaining at the time $t$. ...
0
votes
1answer
37 views

Is there a general formula to compute the number of integer solutions of an equation?

recently, I asked a question concerning the number of solutions of a diophantine equation that used the rounding function. This question, however, dealt with a linear function, and I was wondering if ...
0
votes
0answers
40 views

How to solve $n\ln^{2}(\ln 2^{n}) = g(k)$ for $n$?

I've been trying to find the inverse of an asymptotic function for personal research, and I've gotten it down to: $$n\ln^{2}(\ln 2^{n}) = \exp(\frac{9}{64}\ln^{3}(2^{k})))$$ where $\ln n$ is the ...
0
votes
0answers
49 views

Algebra Revision for iB HL math test -Logarithms, Exponentials and Disciminants

So I'm trying to revise for an upcoming algebra test but there are a few questions in the book that I cant see to find a workaround for; even though I've tried to come up with my own solutions to no ...
0
votes
1answer
79 views

Homework solving exponential equation, logarithmic equation and exponential equation.

I need help with three homework questions.¨ First one: $$\sqrt{3x^2-2x-15}=x+1$$ I don't know how to get the right answer. The answer is supposed to be 4. I get: $$ 3x^2-2x-15=(x+1)^2$$ $$ ...
1
vote
2answers
75 views

Find all values of $x$

Determine all real values of $x$ such that: $$\log_{2}(2^{x-1} + 3^{x+1}) = 2x - \log_{2}(3^x) $$ Let $u = 2^x$ and let $y = 3^x$ For ease, let $\log_{2}$ be represented by just $\log$ so: ...
5
votes
4answers
554 views

Summing reciprocal logs of different bases

I recently took a math test that had the following problem: $$ \frac{1}{\log_{2}50!} + \frac{1}{\log_{3}50!} + \frac{1}{\log_{4}50!} + \dots + \frac{1}{\log_{50}50!} $$ The sum is equal to 1. I ...
0
votes
4answers
39 views

Simplifying a logarithmic expression.

I have: $\log xy + \log 2x^2 - 0.5\log 4y^2$ The unlike terms make it hard to see what can be done? Thanks.
11
votes
1answer
138 views

How do I construct a function $\operatorname{sog}$ such that $\operatorname{sog}\circ\operatorname{sog} = \log$?

Imagine a real-valued semilog function $\DeclareMathOperator{\sog}{sog}\sog$ with the property that $$\sog(\sog(x)) = \log(x)$$ for all real $x>0$. My questions: Does such a function ...
0
votes
0answers
17 views

Why does the sawtooth graph that uses cos(x) instead of sin(x) have a minimum value of -ln(2) when x is a multiple of pi?

So you know how the sawtooth function is $\sum _{n=1}^{\infty}\frac{\sin \left(n\left(x\right)\right)}{n}$, and that the minimum value approaches -2, right? So when I use cos(x) instead of sin(x) ...
2
votes
1answer
46 views

solve $y = \frac{A }{\frac{B}{\ln(y/y_0)} - 1} \frac{1}{x^2}$

I'm trying to express y as a function of x, using the following equation : $$ y = \frac{A }{\frac{B}{\ln(y/y_0)} - 1} \frac{1}{x^2} $$ Can anyone help me ? Thanks ! [Edit] - I originally attempted ...
0
votes
2answers
41 views

If $a=b\log b$, how does $b$ grow asymptotically?

If $a=b\log b$, how does $b$ grow asymptotically in terms of $a$? I think the answer should be $b=\Theta\left(\frac{a}{\log a}\right)$. I tried taking logs to get $\log a=\log b+\log\log b$, but it's ...
0
votes
3answers
47 views

How do we compute this sum?

Given: $$f(x)=\sum_{n=0}^{\infty} \frac{2x^{2n+1}}{2n+1}$$ How do we show that: $$\sum_{n=0}^{\infty} \frac{1}{(2n+1)4^n} = ln3$$ Hints given are that $$f(1/2)=\sum_{n=0}^{\infty} ...
0
votes
1answer
40 views

Natural Logarithm Notation

Perhaps am the only one wondering that... Why the notation of the natural logarithm changes according to the reference is used. So, is the following TRUE? $$\ln{\left| x \right|} = \ln{x} = ...