Questions related to real and complex logarithms.

learn more… | top users | synonyms

1
vote
5answers
46 views

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: $$2=3^{\frac{\ln{2}}{\ln{3}}}$$
0
votes
4answers
82 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 ...
4
votes
3answers
667 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 ...
0
votes
1answer
24 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: ...
0
votes
2answers
38 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. ...
1
vote
1answer
23 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 ...
0
votes
0answers
24 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 ...
0
votes
1answer
40 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: ...
0
votes
5answers
56 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 = ...
1
vote
3answers
68 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?
0
votes
1answer
40 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 ...
2
votes
0answers
184 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 ...
2
votes
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 ...
0
votes
3answers
40 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.
1
vote
0answers
39 views

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 $$ ...
4
votes
1answer
125 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 ...
0
votes
1answer
27 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 ...
0
votes
1answer
42 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?
0
votes
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!
1
vote
1answer
50 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 ...
2
votes
2answers
45 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?
1
vote
1answer
31 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 ...
0
votes
0answers
48 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 ...
1
vote
0answers
25 views

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 ...
3
votes
2answers
88 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 ...
1
vote
3answers
80 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, ...
0
votes
1answer
31 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 ...
0
votes
3answers
51 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 ...
0
votes
1answer
55 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 ...
0
votes
4answers
47 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 ...
0
votes
2answers
50 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 $\ln\left(\frac{a}{b}\right)x + \ln(b) = \ln(c)$ ...
0
votes
2answers
119 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 ...
2
votes
3answers
74 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
1
vote
2answers
77 views

Is the complex square root of $z^2 = \pm z$?

Is $\sqrt{z^2} = \pm z$, for $z$ complex? I think it is, since either $-z$ or $+z$ satisfies the definition $\sqrt{z^2}= e^{\large \frac{1}{2}\log(z)^2}$ but I just wanted to make sure. It's a bit ...
1
vote
0answers
35 views

On the dog-bone contour around [-1,1], what are the arguments of these two lines approaching the real axis from above and below?

I am using a dog-bone contour to integrate around the interval [-1,1]. (-1 and +1 are branch points of the integrand.) I am using the principal branch of log, so I am restricting its argument to ...
1
vote
4answers
65 views

Simultaneous Equations (Stuck on the algebra)

Question: Solve the following simultaneous equations for real values of x and y $$ \left\{ \begin{array}{l} 9^{2x+y} - 9^x \times 3^y = 6 \\ \log_{x+1}(y+3) + \log_{x+1}(y+x+4) = 3 ...
0
votes
1answer
18 views

Given $10^3=1000,10^4=10000,2^{10}=1024,2^{11}=2048,2^{12}=4096,2^{13}=8192$,what are the largest $a$and smallest $b$ such that $a < \log_{10} 2 < b$

If one uses only the information $10^3=1000,10^4=10000,2^{10}=1024,2^{11}=2048,2^{12}=4096,2^{13}=8192$,what are the largest $a$ and smallest $b$ such that one can prove $a < \log_{10} 2 < ...
0
votes
1answer
37 views

Domain of the nested logarithmic function

Find the domain of the definition of the function $$f(x)=\log_{0.3}\left(\log_{0.5}\left(\log_{0.8}\left(x^2-x+1\right)\right)\right)$$ My Try: I assumed $$f_1(x)=x^2-x+1$$ ...
3
votes
0answers
33 views

Branch points and Riemann surfaces (analytic continuation),

Take probably the most typical example: $$f(z) = \sqrt{1-z^2}$$ This function uses the (complex) logarithm to define it: $$e^{\large \frac{1}{2}log(1-z^2)}$$ $$e^{\large \frac{1}{2}[ln|1-z^2| + ...
3
votes
3answers
85 views

How to compute $\int_{0}^{(e-1)^2}{\ln(\sqrt{x}+1)} \,\mathrm dx $?

I have a problem with this integral. $$\int\limits_{0}^{(e-1)^2}\!\! \left({\ln(\sqrt{x}+1)} \right)\,\mathrm dx $$ I applied the substitution method $t = \sqrt{x}+1$, $2t = dx$ I changed integration ...
2
votes
3answers
78 views

For which values of $a\in\mathbb{R} $ the equation $2 \log(x+3)=\log(ax)$ has exactly one root?

I have to investigate the possible roots of the equation according to $a$, i.e. i have to see whether there is only one root, two roots, or no roots and also what their sign is each time. This is from ...
2
votes
3answers
39 views

Divergence of $\sum_{n\geq 2} \frac{1}{\ln^p n}$ for $1<p\leq \infty$ [closed]

Can anyone help me to prove that $(x_n)\notin l_p$ with $x_n=\frac{1}{\ln^p n}$? Suppose $1< p<\infty$.
1
vote
1answer
33 views

How to derive this simple equality?

Let us define $L_i\triangleq \log \left( \dfrac{Prob(x_i=+1) }{ Prob(x_i=-1)} \right)$ $E\{x_i\} \triangleq Prob(x_i=+1)-Prob(x_i=-1)$ I need to show that \begin{equation} E\{x_i\} = ...
8
votes
3answers
232 views

Integral $\int_0^{1/2}\arcsin x\cdot\ln^3x\,dx$

It's a follow-up to my previous question. Can we find an anti-derivative $$\int\arcsin x\cdot\ln^3x\,dx$$ or, at least, evaluate the definite integral $$\int_0^{1/2}\arcsin x\cdot\ln^3x\,dx$$ in a ...
0
votes
2answers
37 views

If $f(x)=\log \left(\cfrac{1+x}{1-x}\right)$ for $-1 < x < 1$,then find $f \left(\cfrac{3x+x^3}{1+3x^2}\right)$ in terms of $f(x)$.

If $f(x)=\log \left(\cfrac{1+x}{1-x}\right)$ for $-1 < x < 1$,then find $f \left(\cfrac{3x+x^3}{1+3x^2}\right)$ in terms of $f(x)$. My Attempt $$f ...
2
votes
1answer
79 views

$c= (a^{x}-b^{x})$ where $a$,$b$ and $c$ are known real constants. Solve for $x$.

I tried taking $\log$ on both side but i ended with $\log(a^{x}-b^{x})$ which is difficult to solve. Does anybody has idea how to solve the above equation for $x$.
1
vote
0answers
25 views

Problem with custom made natural log and power functions

I have made these two functions with the help of posts on math.stackexchange.com. For ln I'm using information gathered from Calculate Logarithms by Hand and for ...
3
votes
2answers
94 views

How can a positive integrand integrate to 0? [duplicate]

I integrated $\dfrac{\log x}{1+x^2}$ from $0$ to infinity with residue calculus and got... $0$. This also agrees with Wolfram Alpha. How can this be? Is it due to the behavior of $\log(x)$ near ...
1
vote
1answer
20 views

Hyperbolic Cosine: System of Equations, Isolate Variables

Background information that may help you answer: Alright, so I'm working on a formula that posits that there are a unique pair of coordinates $(x_1, y_1)$ and $(x_2, y_2) $ on the hyperbolic cosine ...
0
votes
1answer
72 views

Growth rates slower than logarithmic? [closed]

So far, I've been able to determine growth rates using the following limit:$$\lim_{x\to\infty}\frac{f(x)}{g(x)}$$Which, if need be, can be solved with calculus. From this, I deduced that it is very ...