Questions related to real and complex logarithms.

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2
votes
5answers
192 views

Textbook clarification: $\log = \ln$

Textbook reads: All logarithms are natural logarithms: $\log = \ln$. Does this mean $n\log(n) = n\ln(n)$?
1
vote
0answers
43 views

Trajectory With Air Resistance

For a video game, I am trying to calculate the angle needed for a projectile to hit coordinates x,y (both non-zero) with air resistance, i used equations from this site, and derived a function of y ...
0
votes
1answer
26 views

How to seperate out a variable from a log

I have: $$20\ln(1 + r/4) = \ln(4/3)$$ I'm trying to solve for $r$. Now if it was just $\ln(r/4)$, it would be easy: $\ln(r) - \ln(4)$, but in this case with a $1 + $ in front, I'm a little confused ...
17
votes
2answers
431 views

Irrationality of sum of two logarithms

I try to prove that the number $$\log_2 5 +\log_3 5$$ is irrational. But I have no idea how to do it. Any hints are welcome.
0
votes
2answers
55 views

Why is $y{(\log_a(x))} = \log_a{(x^y)}$?

Why is $y{(\log_a(x))} = \log_a{(x^y)}$? I feel like I'm missing something here. Sorry if I put the title wrong..
0
votes
0answers
20 views

Estimation with logarithm

I have to prove: $q$ an integer $\geq 2$, $\tau$ a real number $\geq 2$ and $\sigma \leq 1- \frac{1}{\log q\tau}$, $M\geq 0$. Then $M^{1-\sigma }\leq (q\tau )^{-\sigma }$ and $(M+1)^{-\sigma }\leq ...
1
vote
1answer
39 views

Tricky logarithm problem

I having a problem in this logarithm problem involving modulus- Solve for x |x-1|^((log(x))^2-2log(x))=|x-1|^3 Bases same so powers equal. If I take log x as a then I get the following quadratic- ...
0
votes
0answers
20 views

Logarithm inequality theoretical problem

I just want to ask why is it that the inequality sign reverses when we take antilog of an in equation. I understand it a bit practically but is there a proof? Like if log(f(x))< log(g(x)) Then ...
5
votes
2answers
458 views

How to solve this equation? Can I treat as a quadratic equation?

$$\ln(x+3)+\ln(x-4)=0$$ How to solve this equation? First removing the 'ln' from the equation and after making a quadratic equation and then solve the quadratic equation?
0
votes
1answer
29 views

What is $log( b,a)$ according to google?

I expected that $log(b, a)$ represents $log_ba$. However this is not what google calculates for you if you type that into the search bar. For example, google says $log(4,2) \approx 0.62324929039$. ...
0
votes
2answers
38 views

derivative of $e^{\ln x^2}-3x^7$

$$e^{\ln x^2}-3x^7$$ The first term: $=e^v$ $v=\ln x^2=u^2$ $v\;'=2uu\;'=(2\ln x)\dfrac{1}{x}=\dfrac{2\ln x}{x}$ $\dfrac{e^{\ln x^2}2\ln x}{x} +21x^{-8}$ How do I simplify further? I don't ...
1
vote
3answers
46 views

exact roots of $e^{ax}-x=0$

How can I find the general solution to (not a numerical approximation) $e^{ax}-x=0$ as a function of $a$. I thought maybe something like $\frac{ln(x)}{a}$.
2
votes
3answers
37 views

evaluating derivative of $\log_4(2x^2+1)$

Find the derivative and evaluate at $f\;'(2):$ $$\log_4(2x^2+1)$$ $\log_4(2x^2+1)=y$ $4^y=2x^2+1$ $4^y\ln4 \times y\;'=4x$ $y\;'=\dfrac{4x}{4^y\ln4}\implies \dfrac{4x}{(2x^2+1)\ln4}$ What ...
1
vote
4answers
32 views

Evaluating $\frac{d}{dx}\sqrt[4]{\ln(12-x^2)}$

Find Derivative and evaluate at $x=1$: $$ \frac{d}{dx}\sqrt[4]{\ln(12-x^2)} = (\ln u)^{1/4} $$ $$v=(v)^{1/4} \implies v=\ln\;u, v\;'=\dfrac{1}{u}(u\;')$$ $$y\;'=\frac{1}{4}v^{-3/4}\; \times ...
0
votes
2answers
36 views

How to estimate $\sum_{n=0}^\infty (\log\log2)^n/n! $ from below?

How to show that the following inequality holds: $$\sum_{n=0}^\infty \frac{(\log\log2)^n}{n!}>\frac 35$$ Is it possible to prove this using induction?
5
votes
5answers
138 views

Showing that $e^{-2} < \ln 2$

I have to prove the following inequality: $e^{-2} < \ln2.$ Using Bernoulli's inequality, I showed that $2 \leq e$, and using this result I tried to simplify the inequality by using an upper ...
0
votes
3answers
42 views

Evaluating $\frac{\operatorname d \! \phantom x}{\operatorname d\!x}\frac{4}{\ln(x^2+2)}$

$\dfrac{\operatorname d \! \phantom x}{\operatorname d\!x}\dfrac{4}{\ln(x^2+2)}= \dfrac{4}{\ln u}$ $u=x^2+2$ $u\;'=2x$ $y\;'=\dfrac{4}{\dfrac{1}{u}} \times (u\;') \implies ...
0
votes
3answers
21 views

Derivative of $\frac{d}{dt}\ln(6t^2+9t+12)=$

$\dfrac{d}{dt}\ln(6t^2+9t+12)=$ $y=2\ln(6t)+\ln(9t)+\ln(12)$ $y\;'=2\dfrac{1}{6t}(6)+\dfrac{1}{9t}(9)+0$ $=\dfrac{12}{6t}+\dfrac{9}{9t}=\dfrac{2}{t}+\dfrac{1}{t}$ What am I doing wrong?
9
votes
1answer
190 views

Log integrals IV

It can be determined that the integral \begin{align} \int_{0}^{\pi/2} \frac{x}{\sin(x)} \ln\left(\frac{1+\cos(x) - \sin(x)}{1+\cos(x) + \sin(x)} \right) dx \end{align} has a finite value. Is there an ...
9
votes
1answer
167 views

Integral ${\large\int}_0^1\frac{\ln^2\ln\left(\frac1x\right)}{1+x+x^2}dx$

Gradshteyn & Ryzhik, 7th ed., p. 570, formula 4.325(5) give the following definite integral: ...
1
vote
4answers
58 views

How to evaluate $\lim_{x \to 0} ( \ln(1 - \sin x) + x)/x^2$ without using l'Hôpital?

How to evaluate $$\lim_{x \to 0} \frac{\ln(1 - \sin x) + x} {x^2}$$ without using l'Hôpital? I am not able to substitute the right infinitesimal. Is there a substitute? Background We have yet not ...
-2
votes
1answer
73 views

how to evaluate $ \lim_{x\to 0} \frac{\ln (x)}{1-x} $? [on hold]

i know there is a special limit that is $$\lim_{x\to 0} \frac{\ln(x+1)}{x} = 1$$ but from here i cant determine the value of $$ \lim_{x\to 0} \frac{\ln (x)}{1-x} $$ could someone offer some kind ...
3
votes
3answers
62 views

Logarithmic inequality for a>1

Is $\log_{\sqrt a}(a+1)+\log_{a+1}\sqrt a\ge \sqrt6$ always true for $a>1$? What is the approach? Do we check the first a's and then form a induction hypothesis?
6
votes
2answers
98 views

How to Solve : $ A =\frac{1}{6}\left((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3\right) $

$ A =\frac{1}{6}\left((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3\right).$ Solve for $2^A.$ (no calculators or graphs are permitted) The way I went about solving this problem was using ...
3
votes
3answers
86 views

Approximation of Natural Logarithm using arithmetic.

A friend of mine posed this question to me a couple days ago and it's been bugging me ever since. He told me to take the square root of 5 twenty times, subtract 1 from it, and then multiply it by ...
0
votes
1answer
35 views

How to solve: $x^22^{x+1} + 2^{|x-3|+2} = x^22^{|x-3|+4} + 2^{x-1}$

Any help would be appreciated. :) I tried splitting the equation about $x=3$, but the terms $x^2$ and $2^x$ Together in the equation(s) are troubling me. I don't know why I'm unable to apply the ...
0
votes
2answers
55 views

Step by step (show-your-work) example on how to solve a log problem algebraically

It's been a long time since I have done calculus so if someone could please refresh my memory on the steps to solve the following problem algebraically that would be most appreciated. I am interested ...
0
votes
2answers
29 views

How to solve for $x$ when $a>0$: $log_{a+x}x \le log_ax^2 $ [closed]

After reaching $log_{a+x}x\le 2log_ax$ and concluding $x>0$ I'm not able to solve further. Hint?
2
votes
3answers
53 views

Is there an approximation to the natural log function at large values?

At small values close to $x=1$, you can use taylor expansion for $\ln x$: $$ \ln x = (x-1) - \frac{1}{2}(x-1)^2 + ....$$ Is there any valid expansion or approximation for large values (or at ...
-1
votes
2answers
44 views

solve $\log_3^2(x)-\log_2(x)=2$

The solution for the equation $\log_3^2(x)-\log_2(x)=2$ is a) $s=\{2,-1\}$; b) $s=\{6,-3\}$; c) $s=\{9, 1/3 \}$; d) $s=\{27, 1/9 \}$; e) $s=\{ 1/6 ,12\}$ It was given in a test at school and I ...
1
vote
5answers
61 views

For small $x$, one has $\ln(1+x)=x$?

What does it mean that for small $x$, one has $\ln(1+x)=x$? How can you explain this thing ? Thanks in advance for your reply.
0
votes
1answer
60 views

Log integrals III

The integral \begin{align} J_{m} = \int_{0}^{1} \frac{t^{m}}{1+t} \, \ln(1+t) \, dt \end{align} has the general form \begin{align} J_{m} = (-1)^{m} \left[ A_{m} - B_{m} \, \ln(2) + C_{m} \, ...
1
vote
2answers
37 views

Easy limit calculation $\lim_{n\to +\infty} n(a^{1/n} -1)$

I have to calculate the limit $\lim_{n\to +\infty} n(a^{1/n} -1)$. I found that it tends to $a$ but don't really see how to prove it with one or 2 steps... Can you please help me out ?
0
votes
0answers
20 views

How to calculate multiple logarithms for complex numbers?

How to calculate $\log(\log(\log(w)))$ where $w=u+yi$ is a complex number? I reckon I need to use equation $w=|w|e^{i\phi}$ somehow? Any ideas?
-1
votes
1answer
31 views

What will be the value of k?

I was solving a problem and in the middle of that problem I encountered an equation from which value of k was needed to be figured out. What will be the value of k in terms of n ? Also how to find it ...
0
votes
1answer
13 views

Change formula from EV to Shutter Speed equivalent

I have this formula: $$\mathrm{EV}=\log_2\frac{N^2}t,$$ How can I extract $t$? $t = ?$
2
votes
2answers
96 views

Log base 10 not equal to log while differentiating?

I was looking at sample questions from my textbook and I came across something interesting that I need a little help understanding The question was to find the derivative of: $\log_{10} ...
1
vote
2answers
27 views

Sum Representation of log(1 + x)

$\log(1+x) = \sum_{k=1}^{\infty} \left(\dfrac{x}{1+x}\right)^{k} \dfrac{1}{k} = \sum_{k=1}^{\infty} \left(1 - \dfrac{1}{1+x}\right)^k \dfrac{1}{k}$ Why is this true? The most sum representation of ...
3
votes
2answers
98 views

Log integrals II

By considering the integral \begin{align} I_{\mu} = \int_{0}^{\pi/4} \sin(2\theta) \, \left( \cos(\theta) - \sin(\theta) \right)^{\mu} \, d\theta \end{align} derivatives can be taken with respect to ...
10
votes
1answer
185 views

What is a closed form for ${\large\int}_0^1\frac{\ln^3(1+x)\,\ln^2x}xdx$?

Some time ago I asked How to find $\displaystyle{\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$. Thanks to great effort of several MSE users, we now know that \begin{align} \int_0^1\frac{\ln^3(1+x)\,\ln ...
2
votes
0answers
61 views

Is it possible to solve $k = \frac{x}{\ln(x)}$ for $x$?

Is it possible to solve $k = \frac{x}{\ln(x)}$ for $x$? My suspicion after a fruitless hour of manipulation is that it is not.
6
votes
1answer
159 views

Two integral involving logarithm and trigonometric function [closed]

Evaluate the following integrals $$\int\limits_0^{\frac{\pi }{2}} {{x^3}{{\ln }^2}\left( {\sin x} \right)dx} ,\int\limits_0^{\frac{\pi }{2}} {{x^3}{{\ln }^2}\left( {\cos x} \right)dx} .$$ Can you ...
5
votes
5answers
425 views

L'Hôpital's as $x$ tends to infinity

I'm searching for the explanation to the limit of: $$ \lim\limits_{x\to\infty} x\, \ln\frac{x+1}{x-1}. $$ I know the answer is 2, but I can't seem to get there. The problem is in my textbook under a ...
0
votes
1answer
20 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
0
votes
1answer
32 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 ...
0
votes
1answer
36 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? ...
1
vote
2answers
15 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 ...
0
votes
1answer
188 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 ...
1
vote
4answers
43 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 ...
0
votes
2answers
23 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 ...