Questions related to real and complex logarithms.

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Prob. 7, Chap. 1 in Baby Rudin

Here's problem 7 in the exercises following Chap. 1 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Fix $b > 1$, $y > 0$, and prove that there is a unique real number $x$...
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2answers
134 views

Show that $f(x^a) = a f(x) $ and $f(x y) = f(x) + f(y)$

Let $f(x) = (x-1) \large\prod_\limits{n=1}^{\infty} \dfrac{2}{x^{2^{-n}} + 1} $ For real $x > 0$ it is easy to show that $f(x^2) = 2 f(x)$. Let $a$ be a real number. Question 1 Show that ...
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62 views

Solve for $x$ in $\frac{x}{\ln(x)}=a$. Why does Wolfram alpha use complex numbers here?

Is there any possible way of doing this without using complex numbers? And why are complex numbers used?
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2answers
40 views

Find the value of $\log_{xz}(xy^4z) \times \log_{xy}(xyz^4)$

if $x,y,z \gt 1$ and $x^2=yz$ find the value of $$E=\log_{xz}(xy^4z) \times \log_{xy}(xyz^4)$$ what i did is $$E=\log_{xz}(xy^4z) \times \log_{xy}(xyz^4)=(1+4\log_{xz}y)\times (1+4\log_{xy}z)$$ ...
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2answers
41 views

Change of base proof without reciprocal

I am looking for a proof of the change of base formula without using the reciprocal. I know that: $$log_ax=\frac{log_bx}{log_ba}$$ The proof usually involves taking the reciprocal: $log_ax=y$ ...
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2answers
53 views

Are there other functions of sets $f$ such that they have this property?

$f(A \cap B) = f(A) \cup f(B)$ This function is similar to the $\log_c$ function in that application of it onto a multiplication is equivalent to the summation of its applications. $\log_c(ab) = \...
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0answers
57 views

Three almost-integers of the form $ce^{H_a+H_b}\approx 2^k\pm1$

The approximation $$H_n\approx log(2n+1)$$ http://math.stackexchange.com/a/1602945/134791 suggests that the harmonic number for composite odd numbers might be close to the sum of the harmonic ...
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3answers
23 views

How to express $\log_3(2^x)$ using $\log_{10}$? And how to evaluate $4^{\log_4y}$?

How to express $\log_3(2^x)$ using $\log_{10}$? And how to evaluate $4^{\log_4y}$?
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3answers
93 views

Showing that the series $\sum \log{n}/n^2$ converges.

I aim to show that the series $$\sum \frac{\log{n}}{n^2}$$ converges. I know that the inequality $log(n) < \sqrt{n}$ holds for large $n$. So this give us one way to prove the convergence of $\sum \...
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0answers
17 views

A group with unusual discrete log properties.

Does there exist a group where computing $g^x$ from $g^{a^{x}}$ is easy, computing $g^{a^{x}}$ from $x$ and $g^{a}$ is hard, and computing $x$ from $g^a$ and $g^{a^x}$ is hard. Intuitively I would ...
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1answer
61 views

Solving logarithmic equations without calculator

Hi I am stuck on this question $$ \log_x 10= 5 (\log_{10} x) +4 $$ The answer key gives the solutions $x = 10^{1/5}$ and $x = 1/10$.
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3answers
64 views

Find the value of $3^{\log_4(5)} - 5^{\log_4(3)}$. [closed]

Find the value of $3^{\log_4(5)} - 5^{\log_4(3)}$. Is there any property that can help here?
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1answer
136 views

Divergent function of ratio must be logarithm

Given. Consider two functions $F(t)$ and $r(t,x)$ such that $\lim_{t\to\infty} F(t) = \infty$ and $\lim_{t\to\infty} r(t,x)$ is finite for any $x$. ($x$ and $t$ are always positive in what follows.) ...
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4answers
72 views

Paradox: Summation of natural logarithms

Consider the expression : $$\sum_{i=1}^{\infty}\ln(i+2)-\ln(i+4)$$ If one evaluates it out, one gets $$\ln(\frac{3\times4\times5\times6\times...}{5\times6\times7\times8\times...})=\ln(12)$$ That ...
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1answer
30 views

Why does $\ln(1+\frac{3}{n^2}+o(\frac{1}{n^2}))=\frac{3}{n^2}+o(\frac{1}{n^2})$?

In order to show that a series converges, I want to show that $\sum\ln(\frac{v_{n+1}}{v_n})$ Which led me to the following first part of the equation, but I didn't achieved to solve it so I looked in ...
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2answers
67 views

I would like to calculate this limit: $ \lim_{n \to \infty}(n^2+1)\cdot(\ln(n^2-4)-2\ln(n)) $

I would like to calculate this limit: $$ \lim_{n \to \infty}(n^2+1)\cdot(\ln(n^2-4)-2\ln(n)) $$ but I am a bit lost on how to tackle the logarithm. Any help would be greatly appreciated.
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4answers
62 views

Is $\log(n!) \in\Theta(n \log n)$ [duplicate]

Is $\log(n!) \in\Theta(n \log n)$? I know it is $O(n \log n)$ because $\log(n!) \leq \log(n^n)$ which is the same as $\log(n!) \leq n \log n$. But how can I show it is also $\Omega(n \log n)$?
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5answers
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logarithms properties

I know it's very easy and naive, but apparently I cannot understand the following equation. Can you please prove it? Thanks in advance. The equation is: $$2=3^{\frac{\ln{2}}{\ln{3}}}$$
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4answers
83 views

How do you solve the equation $0.5^x = 2^x + 3$?

I need help with the following problem: $0.5^x = 2^x + 3$ I know the answer is -1.72, but I have to explain step by step how to solve it and I'm not sure how. I know you're supposed to take the log ...
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3answers
686 views

Confusion regarding $\log(x)$ and $\ln(x)$

I was solving an integral and I encountered in some question $$\displaystyle \int_{2}^{4}\frac{1}{x} \, \mathrm dx$$ I know its integration is $\log(x)$. But my answer comes correct when I use $\ln(...
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1answer
33 views

Logarithm of complex matrix

For invertible matrix $A$, we have $\log(\det A) = \mathrm{tr}(\log A)$ due to a corollary of Jacobi's formula. What if we had the argument $iA$ instead? Would the above relation still hold? Edit: ...
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39 views

How to prove $cn < n^{\log_{2}n}$

How do you prove that for any given $c$, there exists an $n$ such that $$cn < n^{\log_{2}n}$$ ? I know that I have to write $n$ in terms of $c$, but I'm having trouble with the log in the exponent.
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1answer
26 views

Derivative of the inverse of exponential function a^x, with a>0 and a≠1

While studying exponential functions, I understood that $$\frac{d}{dx}a^x=(\ln a)a^x.$$ I also learned previously that if $g(x)$ is the inverse of $f(x)$, then the derivative of $g(x)$ and the ...
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0answers
35 views

An…almost inverse Mellin Transform?

$$\log x= \frac{1}{2\pi i} \int_{\gamma-i\infty}^{\gamma+i\infty} \frac{\Gamma^2(-n)\Gamma(n+1)}{(x-1)^n\Gamma(-n+1)}\text{d}n$$ I thought that this was an Inverse Mellin Transform, but ultimately ...
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1answer
45 views

Need to show the equality of logarithm.

To show: $ \lim_{n \rightarrow \infty} 2n ( a^{\frac{1}{2n}} -1)=log(a) $ for a>0. By definiton: $$ e^{x}=\lim_{m \rightarrow \infty}(1+ \frac{x}{xm} )^{xm} =: a $$ Now take the log of both sides: ...
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5answers
60 views

What is the Inverse function of $y = 10^{-x}$? Steps are appreciated.

What is the inverse of $y = 10^{-x}$? These are my steps for the problem. Step 1 $y = 10^{-x}$. Step 2 $x = 10^{-y}$ by inverse substitution. Step 3 $10^y(x) = 1$. Step 4 $10^y = \frac{...
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3answers
82 views

Which is asymptotically larger $n^2 \log(n)$ or $n (\log(n))^{10}$?

Which is asymptotically larger $n^2 \log(n)$ or $n ( \log(n))^{10}$? I have tried by plugging in the values and $n^2 \log(n)$ turns out to be bigger. How can this be done analytically?
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1answer
43 views

Can I take an exponent out of a sum?

For example, assuming we had a sum: $$\sum_{n=1}^m n^b \quad m,b\in\mathbb{N}$$ Is there any way to take the $b$ out of the sum? I tried taking the $\log_n$ of every value, add them together then ...
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1answer
320 views

Why is $e$ close to $H_8$, closer to $H_8\left(1+\frac{1}{80^2}\right)$ and even closer to $\gamma+log\left(\frac{17}{2}\right) +\frac{1}{10^3}$?

The eighth harmonic number happens to be close to $e$. $$e\approx2.71(8)$$ $$H_8=\sum_{k=1}^8 \frac{1}{k}=\frac{761}{280}\approx2.71(7)$$ This leads to the almost-integer $$\frac{e}{H_8}\approx1....
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4answers
50 views

Solve $\log_{1/3}(x^2-3x+3)≥0$

I want to solve $$\log_{1/3}(x^2-3x+3)≥0$$ Now I know the result is: $x ∈ <1;2>$, but i am not sure how to get it. My thoughts: $\frac{1}{3}$ to the power of positive number $= (x^2-3x+3)$, now ...
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3answers
41 views

Solve $\log_{1/4}{x}=\frac{3}{2}$

I want to solve $$\log_{1/4}{x}=\frac{3}{2}$$ Now I know the result is: $$\frac{1}{8}$$ but I am not sure how to get it. Any help would be greatly appreciated.
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0answers
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A binary BBP-type formula for log(23)

This question is related to Is there a binary spigot algorithm for log(23) or log(89)? by Dan Brumleve. A binary BBP-type formula is a convergent series formula of the type $$ \alpha=\sum_{k=0}^\...
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1answer
156 views

A $\log \Gamma $ identity: Where does it come from?

$$\log \Gamma (n)=n\log n -n +\frac{1}{2} \log \frac{2\pi}{n}+\int_0^\infty \frac{2\arctan (\frac{x}{n})}{e^{2\pi x}-1} \,\mathrm{d}x$$ Is an identity that is derived from using Sterling's ...
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1answer
36 views

Solving logarithmic equation, different bases

What number do I need to multiply both sides with? I have worked for an hour on this but it is the first time I am using this website so it is impossible for me to write what I have already done. If ...
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1answer
52 views

Integrate $\int(\log(\sin x \cos x))^n dx$ with hypergeometric function form

Evaluate $$\int({\log(\sin x\cos x)})^{n} \, \mathrm{d}x$$ with result in hypergeometric function form Could anyone help me with that?
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2answers
26 views

Find number of digits of a number in another base

How can I solve this question: Given that $\log 3$ is about $0.48$, approximately how many digits are in the number $10^{150}$ if it were written in base $3$. Thanks!
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1answer
116 views

Finding Stationary Points of Natural Log Function

$$ f(x) = x - 2\ln(x^2 + 3) $$ I started by using the chain rule on $x^2 + 3$ which gives me $\frac{2x}{x^2} + 3$. At this point I tried to multiple $\frac{2x}{x^2} + 3$ by $x - 2$ - is this correct?...
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2answers
51 views

Rejecting a solution.

Why does it for $x^2=9$ we get two solutions, while if we use the "log both sides" property the negative solution is rejected? which method is true and why?
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1answer
32 views

Logarithmic equation with variable both “free” and in logarithm

I am trying to calculate an area bordered by two functions and in the process I need to solve this equation: $$e^{-10x}=-2x+1$$ I make it into a non-exponential form: $$-10x=ln(-2x+1)$$ And now I am ...
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0answers
73 views

integral involving error function (erf)

Does anybody know if a closed form of this integral exist? $\int \mbox{erf}(x) \ln(\mbox{erf}(x)) \Bbb dx$ where erf is so called error function. In case there is no closed form solution. Is it ...
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0answers
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Derivation for Napier Logarithm

Something new that I came across called the Napier Logarithm. I was reading the book “Computing: A Historical and Technical Perspective”, and the book says that $$Nap.log x = 10^7(\log_e(10^7/x))$$ I ...
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2answers
141 views

What is the integral of log(z) over the unit circle?

I tried three times, but in the end I am concluding that it equals infinity, after parametrizing, making a substitution and integrating directly (since the Residue Theorem is not applicable, because ...
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3answers
155 views

Better to memorize logarithm rules? And how?

Do people good at math totally memorize these logarithm rules below? If so, are there good mnemonics for this? I'm bad at math and I only memorize these rules really vaguely by rote, thus when needed, ...
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1answer
33 views

Showing that $\log{\log^d{3n}} = O(\log{\log^d{n}})$

I'm trying to show this: $$\log{\log^d{3n}} \leq q\cdot \log{\log^d{n}} \;\;\exists\, q,k > 0,\forall n>k, \text{where } d \text{ is a constant} > 0$$ This is what I have so far $$\...
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3answers
52 views

Logarithmic function and $x$

Hi I was wondering why in a logarithm $x$ cannot be a negative number, since for the inverse graph I drew the $x$ values are only positive. In the question it asks why the first four points of the ...
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1answer
77 views

Determine equation from graph

Background: I'm working on a script to read/parse a file generated by a piece of software I use to create music mixes. One aspect I'm having difficulty with is translating the volume value from it's ...
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4answers
52 views

Limit with logarithms

Can you please help to calculate the following limit $$\lim_{x \to \infty} \left(\frac{\ln(x\ln(\gamma) )}{\ln(x\ln(\gamma)+\ln(\frac{\gamma-1}{\gamma}))}\right)^{1/2},$$ where $\gamma >10$ is a ...
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1answer
58 views

Solving for $x$ using $\ln$

I have an equation as follows: $\ln(a)x + \ln(b) (1 - x) = \ln(c)$ I'd like to solve for $x$, but I think what I've come up with is wrong $\ln\left(\frac{a}{b}\right)x + \ln(b) = \ln(c)$ $\ln\left(...
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2answers
136 views

Find the maximum possible value of $8(27)^{\log_{6}x}+27(8)^{\log_{6}x}-x^3$,where $x>0$

Find the maximum possible value of $8(27)^{\log_{6}x}+27(8)^{\log_{6}x}-x^3$,where $x>0$ Let $P(x)=8(27)^{\log_{6}x}+27(8)^{\log_{6}x}-x^3$ By using $AM-GM$ inequality on the first two terms ...
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3answers
76 views

How do I evaluate the sum $\sum_{k=1}^\infty\left(\ln\big(1+\frac{1}{k+a}\right)-\ln\left(1+\frac{1}{k+b}\big)\right)$ [closed]

How do I evaluate the sum $$\sum_{k=1}^\infty\left(\ln\Big(1+\frac{1}{k+a} \Big)-\ln\Big(1+\frac{1}{k+b}\Big)\right)$$ where $0 <a<b<1$? Hints will be appreciated Thanks