Questions related to real and complex logarithms.

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1answer
33 views

rules for evaluating powers of logarithms

What rules are we using to show that $3^{-s}=\frac{1}{2}$ if $s=\frac{\log 2}{\log 3}$ I cannot understand how you can raise a number to a logarithm divided by a logarithm
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1answer
33 views

What is the value of x? Related to Indices.

Just some days ago I appeared for a maths exam. In that exam there was a question related to Indices which I was not able to solve. After the exam I even tried solving it in the home next 2 days but ...
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1answer
41 views

Is this proof with logarithmic exponentials correct?

I was unsure of this proof and some of the log rules I applied, could you check my proof and tell me if this proof is correct and if not, then what specifically is incorrect about the proof? $\...
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vote
2answers
18 views

Which conditions imply $\sup_n |\ln x_n| < \infty$?

I want to find conditions which imply that $\sup_n|\ln x_n| < \infty$. Intuitively I think that $\inf_n x_n > 0$ and $\sup_n x_n < \infty$ should be enough, but I don't know how to write it ...
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1answer
8k views

Convert a linear scale to a logarithmic scale

Given a number n, how would I convert this number into a logarithmic scale? My logarithmic scale would range from 0 to 255 (I'm working with RGB colours), and I ...
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4answers
71 views

Criterion to satisfy Rolle's Theorem.

$f(x) = \begin{cases} x^a\log x, & \text{if $x \neq 0$,} \\[2ex] 0, & \text{if $x=0$. } \end{cases} $ What should be the value of $a$ so that f satisfies Rolle's theorem in [0,1] ?? What I ...
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2answers
97 views

Logarithm Equation+ Modulus function

Please help me in answering the following question Find the number of real values of $x$ satisfying the equation: $$\Large \left| 3 -x \right|^{ \log_7(x^2) - 7\log_x (49)} = (3-x)^3$$ I am able to ...
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3answers
23 views

Solving for $x$ using exponential log laws

For $\log_2(x) + 2\log_2(x-1) = 2 + \log_2(2x+1)$ I moved all the $x$ to left side, used got rid of log and got $x-(x-1)^2 - (x+1) = 4$ Simplyifing I get $x^2-2x=4$ The answer should be $x = ...
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2answers
93 views

how to solve $\log{x}=cx^4$ for $x$

I was wondering if there is a general solution for this form of equations: $$\log{x}=cx^4$$ Tried: $$ x = e^{cx^4}\\ xe^{-cx^4}=1$$
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votes
2answers
167 views

Log integrals I

In this example the value of the integral \begin{align} I_{3} = \int_{0}^{1} \frac{\ln^{3}(1+x)}{x} \, dx \end{align} was derived. The purpose of this question is to determine the value of the more ...
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vote
1answer
89 views

Proving n(log(n)) is O(log(n!))

I want to prove $n(\log(n)) \in O(\log(n!))$. I don't really understand how to prove this statement. From the definition, we would have that: $\exists c > 0, \exists N$, so that $\forall n \geq N, ...
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1answer
187 views

How can I approximate the logarithm of the sum?

Consider $\alpha = \log a$ and $\beta = \log b$, $b>a$. Are there formulas for approximating $\gamma = \log (a+b)$? What about $\theta = \log (a-b)$? If it makes it easier, assume that $|\alpha| \...
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2answers
502 views

Equation with logarithms and absolute value

I have this equation: $$ \ln\frac{2-|y-1|}{1-|y|} = \ln x $$ which becomes $$ \ln(2-|y-1|)-\ln(1-|y|) = \ln x. $$ Can the first term in LHS be written as $\ln(2)-\ln(|y-1|)\implies\ln(2)-\ln(|y|)-\...
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votes
3answers
75 views

How to solve $10^{x^2+x}+\log{x} = 10^{x+1}$?

In one of my recent exam, I was ask to solve this: $$ 10^{x^2+x}+\log{x} = 10^{x+1} $$ My attempt to solve it was: $$ 10^{x^2+x}+\log{x} = 10^{x+1} \\ \log{x}=10^{x+1}-10^{x^2+x} \\ \...
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3answers
67 views

Logarithmic inequality

We're working on logs and exponents in class, but I've never dealt with a situation where the equation isn't an equation but an inequality. Any help on how to solve this?! Thanks! $$7^{\sqrt{2x}}>...
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2answers
37 views

Natural logarithm problem

I'm kind of confused on how to solve this problem! Any guidance/advice would be appreciated. Thanks! $e^{−9}e^{−2}e^{9}$
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4answers
241 views

A closed form for $\int_0^1 \frac{\left(\log (1+x)\right)^3}{x}dx$?

I would like some help to find a closed form for the following integral:$$\int_0^1 \frac{\left(\log (1+x)\right)^3}{x}dx $$ I was told it could be calculated in a closed form. I've already proved that ...
3
votes
1answer
134 views

Intuitive understanding of logarithms

I know logarithms are supposed to be the inverse of exponential functions, and while this makes sense, it seems to me that a more intuitive and significant property is $$\log (ab) = \log(a)+\log(b)$$ ...
0
votes
3answers
120 views

Find the product of positive roots of equation $\sqrt{2008}\,x^{\log_{2008}x}=x^2$

Problem : Find the product of positive roots of equation $\sqrt{2008}\, x^{\log_{2008}x}=x^2$ Solution : The given equation can be written as $\sqrt{2008} \, x^{\log_{2008}x}=x^2 $ $\implies\...
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votes
5answers
179 views

How to solve this exponential equation? $2^{2x}3^x=4^{3x+1}$.

I haven't been able to find the correct answer to this exponential equation: $$\eqalign{ 2^{2x}3^x&=4^{3x+1}\\ 2^{2x} 3^x &= 2^2 \times 2^x \times 3^x\\ 4^{3x+1} &= 4^3 \times 4^x \times ...
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4answers
2k views

How to solve this logarithmic equation whose expressions have different bases?

I have been trying to solve the following equation for a while and i can't seem to figure it out, your help would be greatly appreciated. Here is the equation: $3^x$=$5^{x-1}$
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2answers
62 views

How to find the number of solutions of equation $x^n - a^x = 0$?

I have to find the number of solutions of the equation $x^4 - 5^x = 0$ Since it is only asked to find the number of solutions and not the exact solution, what is the best way to approach such ...
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votes
1answer
35 views

Prove $\frac{\ln(x)}{x} \le \log_{10} 2$

How would I go about proving the following inequality? $\frac{\ln(x)}{x} \le \log_{10} 2$ Is there an algebraic solution and if not, why?
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3answers
55 views

Prove $(\log{n})^2\leq 2^n$ by induction

I've trying to solve this for quite a while now, but not being able to finish the proof. Prove using induction that $(\log{n})^2\leq 2^n$
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vote
1answer
74 views

Why is the derivative of $\ln |x|$ equal to $1/x$?

In a complex analysis book, I read that $f(z) = 1/z$ doesn't have a primitive on $\mathbb{C}\setminus\{0\}$. The reason given used the much stronger fact that if any $g : \mathbb{C} \to \mathbb{C}$...
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1answer
55 views

Logarithms in Calculators?

I have no idea how to do logarithms, or even what they are, but our class recently received an extra credit problem pertaining to one. This helps me with EXACTLY what I want to do, but I have no idea ...
14
votes
4answers
445 views

Evaluate $\int^1_0 \log^2(1-x) \log^2(x) \, dx$

I have no idea where to even start. WolframAlpha cant compute it either. $$\int^1_0 \log^2(1-x) \log^2(x) \, dx$$ I think it can be done with series, but I am not sure, can someone help a little so ...
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1answer
756 views

Stuck on derivative of logarithm of sum of exponentials

let's say that I need to calculate the following expression: $$ \frac{\partial\mathrm{log}(\mathrm{exp}(w_1 * x_1 + b_1) + \mathrm{exp}(w_2 * x_2 + b_2))}{\partial w_1} $$ How do I start? The rules ...
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2answers
70 views

stuck on logarithm of derivative of sum $\frac{\partial\mathrm{log}(a+b)}{\partial a}$

I need to evaluate an expression similar to the following: $\frac{\partial\mathrm{log}(a+b)}{\partial a}$ At this point I don't know how to proceed. $b$ is a constant so there should be some way to ...
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2answers
39 views

Finding the value of an expression with logarithms

Given that $\log_{b}a=0.74$ and $\log_{b}(a-1)=0.65$ find the value of the following expression: $$\log_{b}(a^{4}-1)-2\log_{b}(a^{2}+1)+\log_{b}(a^{3}+a)-\log_{b}(a+1)$$ I tried using log laws to ...
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2answers
78 views

Explain how to get the right solution of y $dy/dx=y$

When solving the following equation to find y as a function of x: \begin{equation} dy/dx=y \end{equation} First I divide both sides by $y$ and multiply both sides by $dx$: $dy/y=dx$ Then I ...
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0answers
82 views

Evaluating a limit involving fractions and logarithms

I am trying to evaluate the following limit. Let $0 < \alpha < \infty$. Then $\begin{align*} \lim_{k \to \infty} \frac{\log(k)}{\log\left[(k+1)^\alpha(k)^{\alpha}\right]-\log\left[(k+1)^{\alpha}...
2
votes
3answers
67 views

Proof for $\log\frac{2n+1}{n+1}<\frac{1}{n+1}+\frac{1}{n+2}+…+\frac{1}{2n}<\log 2$

How can I prove that $\displaystyle \log\frac{2n+1}{n+1}<\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}<\log2$ using $\displaystyle \frac{x}{1+x}<\log(1+x)<x$
1
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1answer
68 views

Check my answer, $\lim_{n \to \infty}\frac{2^{\sqrt{\log (\log(n))}}}{(\log(\log(n)))^2\sqrt n}$

I would like someone to review my solution to this limit, the result (at least to me) is quite surprising. Assume there is a limit and it is $L$, then $$\lim_{n \to \infty}\frac{2^{\sqrt{\log (\log(...
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votes
3answers
580 views

What is the meaning of $\log^2n$ and how should it be read in word form?

$\log^2n$ is what I need assistance with. How is this read in word form? What exactly does this mean? No matter how much I read about logarithms, they still seem new to me.
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votes
2answers
437 views

Prove that a logarithm is irrational [duplicate]

I’m stuck with the following problem: Prove that $\log_{2} 3 \in \mathbb{R} - \mathbb{Q} $ . Thanks in advance!
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3answers
114 views

Show that a certain entire function is identically zero

Let $f(z)$ be an entire function satisfying $$ \left|\,f\left(\frac{1}{\log(n+2)}\right)\right|<\frac{1}{n}, \quad\text{for all $n\in \mathbb N$}$$ for $n\in \mathbb N$. Show that $f(z)\equiv 0$. ...
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1answer
170 views

A closed form for $\sum_{n=1}^{\infty}(-1)^{n-1}\arctan\left(\frac{1}{n}\right)\ln(n^2+1) $

This is another 'arctanlog' series: $$ S=\sum_{n=1}^{\infty}(-1)^{n-1}\arctan\left(\frac{1}{n}\right)\ln(n^2+1) $$ Maybe differentiating with respect to some parameter could be of interest. What ...
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2answers
92 views

Any power of logarithm is $O(N)$

This is more of a computer science question but it uses calculus and proof techniques so I think it might be more appropriate here. Basically, how do I prove that, for any constant $K \geq 1$, ...
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1answer
93 views

Cosine of the Natural Logarithm - Series Expansion

I am interested in a computable series expansion of the following equation: $f(n) = \cos(\log(n))$ Specifically, I am interested in real values of $n$ where $n>1$. From basic series definitions ...
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1answer
105 views

What is $\int_{\lvert z\rvert=1} \mathrm{Log}(z)\,dz$?

What is $\int_{\lvert z\rvert=1} \mathrm{Log}(z)\,dz$? By letting $z = \mathrm{e}^{it}$, we get $$\int_{\lvert z\rvert=1} \mathrm{Log}(z)\,dz = \int_0^{2\pi} \mathrm{Log}(\mathrm{e}^{it}) i\...
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2answers
54 views

why is $\displaystyle \frac{\log(\sin x)}{\log(x)}$ $\quad\frac{\infty}{\infty}$ form as $x\to 0$?

In this question $\displaystyle\frac{\log(\sin x)}{\log x}$ is taken as $\displaystyle\frac{\infty}{\infty}$ indeterminate form. But $\log(0)$ is not defined so how can L'Hospital's rule can be used?
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1answer
88 views

How logarithms affect given condition

I am working with long productcs of probabilies and in order of avoinding underflow I am using the addition of (negative) logarithms. P(A) =-log(P(a1) + -log(P(a2)+.... In the end I get a positiv ...
0
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1answer
20 views

Logarithm Via Multiplication By Some Function

This question may be totally rubbish, I'm not sure... Basically, I'm wondering if I have some expression, say, $$f(z)A(z)$$ where $A(z)$ is some arbitraryly changing function in $z$ totally out of my ...
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votes
1answer
83 views

is there any solution for $x^2 +x + 2 = e^x$ by using algebra?

I know this can be solved by numerical methods but I would like to know whether this can be solved using logs or something similar. Thanks
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1answer
87 views

Can $x^2 +x + 2 = 10^x$ be solved using algebra?

I know this can be solved by numerical methods but I would like to know whether this can be solved using logs or something similar. Thanks
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3answers
66 views

Proof of the fact that $\ln(a) = f '(0)$ for $f(x) = a^x$?

Looking over notes from class today and wanted to know if there is any type of proof for the fact that $\ln(a) = \lim_{h\to0}(a^h-1)/h$, which is just $f '(0)$ for any function of the form $f(x) = a^x$...
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2answers
365 views

Integral ${\large\int}_0^1\left(-\frac{\operatorname{li} x}x\right)^adx$

Let $\operatorname{li} x$ denote the logarithmic integral $$\operatorname{li} x=\int_0^x\frac{dt}{\ln t}.$$ Consider the following parameterized integral: $$I(a)=\int_0^1\left(-\frac{\operatorname{li} ...
3
votes
1answer
217 views

Calculus integral evaluation using substitution

I have to find this integral: Evaluate the integral using an appropriate substitution $$\int\dfrac{8e^x+7e^{-x}}{8e^x-7e^{-x}}\mathrm dx.$$ I've tried my solution $\ln\Big[15\cdot \sinh(x) + \...
2
votes
2answers
68 views

Logarithmic function with strange bases

Given $\log_{4n} 40\sqrt{3} = \log_{3n} 45$, find $n$. I have rewritten $\log_{3n} 45$ as $\dfrac{\log_{4n}45}{\log_{4n}3n}$ and multiplied to get $\log_{4n} 40\sqrt{3}\cdot\log_{4n}3n = \log_{4n} ...