Questions related to real and complex logarithms.

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Simplifying logs?

Would $\log_2 (n+1)$ simplify to $\mathcal{O}(\log_2 n)$? I wasn't sure if this was valid since logs aren't distributive and I couldn't find a constant $c$ relating the expressions. If this turns ...
2
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4answers
108 views

How to prove $\frac{4^{1/\log_4(3/4)}}{3^{1/\log_3(3/4)}} = \frac{1}{12}\ ?$

How could we prove that $$ \frac{4^{1/\log_4(3/4)}}{3^{1/\log_3(3/4)}} = \frac{1}{12}\ ?$$ I have reduced it the form $$\frac{4^{\ln(4)/\ln(3/4)}}{3^{\ln(3)/\ln(3/4)}}$$ I am not sure what to do ...
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2answers
338 views

Simplifying the expression of exponential and logarithms

I want to simplify the following expression. $$Y=\text{Bottom} + \frac{\text{Top}-\text{Bottom}}{1+10^{((\log EC50-X))}}$$ $\log$ is base of $10$. Some may know that it's a dose response curve, and ...
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3answers
70 views

Find $n$ in $n \log_2 n = c$

I'm trying to find the value for $n$ in the following equation. $$n \log_2 n = c$$ what is $n$? thanks, Tim
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4answers
181 views

How many times do these curves intersect?

When the curves $y=\log_{10}x$ and $y=x-1$ are drawn in the $xy$ plane, how many times do they intersect? To find intersection points eq.1 = eq. 2 $$\begin{align*} \log_{10}x &= x-1\\ 10^{x - 1} ...
17
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1answer
310 views

Approximation of $\log(x)$ as a linear combination of $\log(2)$ and $\log(3)$

I wonder if it's possible to approximate $\log(n)$, n integer, by using a linear combination of $\log(2)$ and $\log(3)$. More formally, given integer $n$ and and real $\epsilon>0$, is it always ...
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3answers
175 views

Summation of a series.

I encountered this problem in Physics before i knew about a thing called Taylor Polynomials My problem was that i had to sum this series : ...
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1answer
103 views

Solving (or estimating) $x$ in $\tau=\log_x\left(\frac{x+1}{2}\right)$

How would one find a real value for $x$ that satisfies $$\tau=\log_x\left(\frac{x+1}{2}\right),$$ given $0 < \tau < 1$ and $\tau \neq \frac{1}{2}$ (PS I'm not that good with math, so if this ...
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2answers
295 views

Why is $\log_{-2}{4}$ complex?

With the logarithm being the inverse of the exponential function, it follows that $\log_{-2}{4}$ should equal $2$, since $(-2)^2=4$. The change of base law, however, implies that ...
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1answer
160 views

Why does $\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$

I read on Wikipedia that $$\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$$ to a high degree of accuracy. Why is this true? Is this pure coincidence or is there some mathematical ...
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1answer
135 views

Undefined natural logarithm?

The natural logarithm is the logarithm to the base $e$, where $e$ is an irrational and transcendental constant. $$e=\lim_{n\to \infty}\left(1+\frac {1}{n}\right)^n.$$ $$\ln a=\log_{e} a.$$ I know ...
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2answers
314 views

How can we solve: $\sqrt{x} - \ln(x) -1 = 0$?

How could we solve $$\sqrt{x} - \ln(x) -1 = 0$$ without using Mathematica? Obviously a solution is $x = 1$, but what are the other exact solutions? This question is inspired by my first question How ...
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2answers
527 views

How can we solve: $\sqrt{x} + \ln(x) -1 = 0$?

How could we solve $$\sqrt{x} + \ln(x) -1 = 0$$ without using Mathematica? Obviously a solution is $x = 1$, but what are the other exact solutions?
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1answer
99 views

equivalance between two equations with log and exponential

$d > \sigma$ ... (1) $\exp^{-(\frac{d^2}{2\sigma^2})} < 10^{-0.5}$ ... (2) Is (1) <==> (2) true ? EDIT: > replaced by < in the 2nd expression
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2answers
251 views

The Fermat prime 257 and binomial sum $\sum_{n=0}^\infty \frac{(-1)^n}{\binom {8n}{4n}}$?

We have, $\begin{aligned} \sum_{n=0}^\infty \frac{(-1)^n}{\binom n{n/2}} &= \frac{4}{27}(9-\pi\sqrt{3}\,)\\[2.5mm] \sum_{n=0}^\infty \frac{(-1)^n}{\binom {2n}n} &= \frac{4}{5} - ...
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1answer
100 views

How do you solve this equation: $5x^{\frac{1}{x}}=3$

I've been at it for a while but I can't get it. Can anyone help?
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2answers
136 views

Is $2^{\log_2(-5)}$ defined?

As far as I know for $\log_2 x$ to be defined $x$ must be higher than 0. However when I enter $2^{\log_2(-5)}$ into wolframalpha it gives result $-5$. Is it mistake?
4
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2answers
763 views

What does $\log^{2}{x}$ mean?

What is it used for and why doesn't it equal $\large\log{x^2}$?
2
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1answer
103 views

Logarithms - Find the solution of $\ln(x^2+1) = \ln(x) + 2$, how to isolate $x$ in a meaningful way?

When solving $\ln(x^2+1) = \ln(x) + 2$ I'm getting stuck at $e^2 = \dfrac{x^2+1}{x}$ How do I isolate $x$?
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3answers
89 views

Big-$\mathcal{O}$ bounding of sums of logarithmic functions

I am reading a text which states that $$\sum \limits_{n \leq X} \left(\log X - \log n \right) = \mathcal{O}(X)$$ I can't quite see why this is true, though I can certainly believe it. Could anyone ...
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1answer
71 views

Is there any word for a “regular scale”, as opposed to a “logarithmic scale”?

We all know what a "logarithmic scale" means. (It basically means that the distance between 1 and 10 is the same as the distance between 10 and 100, as shown on the figure.) However, what is the word ...
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4answers
150 views

logarithmic find value N.

$\log N=\frac{1}{2}(\log24-\log0.375-6\log3)$ find value N. I did it below step $\log N=\frac{1}{2}(\log64-6\log3)$ $\log N=\frac{1}{2}(\log0.877)$ I don't know how continue further ...
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1answer
331 views

Sketching Logarithmic with labeling intercepts and asymptotes

$y=-2\log(x-2)$ asymptotes $x=2$ lets $x=0$ $y=2\log(0-2)$ $y=2\log(-2)$ lets $y=0$ $0=2\log(x-2)$ I think but I don't know how to draw the graph. The answer of the graph is ...
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1answer
86 views

Solving for $x$ in a log equation

Given $(\log_3 x)^3 = 9 \log x$, solve for $x$. Here is what I have so far: $$(\log_3 x)^3 = \frac{9\log_3 x}{\log_3 10}$$ $$let a = \log_3 x$$ $$a^3=\frac{9a}{\log_3 10}$$ $$a^3-\frac{9a}{\log_3 10} ...
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2answers
83 views

Logarithms - Get the second solution to a logarithm equation

Here's the original equation: $\ln(11x-10) + \Big(\ln(11x-10)\Big)^2$ = 6 I've managed to obtain one solution: $x = \frac{e^2 + 10}{11}$ through those steps: ...
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1answer
263 views

Why is $10\frac{\exp(\pi)-\log 3}{\log 2}$ almost an integer?

I read that $$10\frac{\exp(\pi)-\log 3}{\log 2} =318.000000033252\dots \approx 318$$ Is this simply a coincidence or can this somehow be explained?
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3answers
291 views

Logarithms - How to handle factors within a difference or sum term of logarithms?

Edit: I just rewrote the whole question, to make it clear, what I'm looking for. Originally I asked about solving $16^{x+1} = 4^{x+3}$, then corrected it to $16^{x+2} = 4^{x+3}$. But what I really ...
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4answers
66 views

Simplifing logarithmic equation

I have the result of a differential equation to be: $$\ln(x+3)=3\ln(t+2)+C$$ I want this to be as simplified as possible. Can it be proceeded like: $$e^{(x+3)}=3e^{(t+2)+C}$$ I am not sure about ...
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2answers
2k views

Compound interest - how to solve this with logarithms & geometric series?

I could use some help with the following: Jacques is saving for a new car which will cost 29000 dollars. He saves by putting 400 dollars a month into a savings account which gives 0.1% interest ...
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2answers
286 views

How to compare logs without a calculator

I have multiple different log sums that I need to evaluate. How would I calculate the following without using a calculator or log tables?
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1answer
86 views

What is the $\sum\limits_{i=0}^{\ (\log_2(n))-1)}\frac{n}{2^i}$?

What is the value of the following sum: $$\sum\limits_{i=0}^{\ (\log_2(n))-1)}\frac{n}{2^i}$$ Can you show how to go about arriving at the answer?
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1answer
88 views

A logarithmic equation?

$$\mathcal{O} \left(3^{\log_2(n)} \right) = \mathcal{O} \left(n^{\log_2(3)} \right)$$ Does anyone have any idea how the right side was arrived at? (The $\mathcal{O}$ is Big-$\mathcal{O}$ notation)
2
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3answers
273 views

Simplifying Logarithmic Expression

Compute: $$\frac{1-\log_a^{3}{b} }{(\log_a b+\log_b a+1)\log_a\frac{a}{b}}$$ I tried to expand it : $$\frac{1-\log_a^{3}{b} }{(\log_a b+\log_b a+1)\log_a\frac{a}{b}}$$ ...
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2answers
85 views

Trouble manipulating a log expression

This question sort of follows on from question Functions with logarithmic integrals. The book presents an example of integrating a function whose integral is logarithmic: $$\int \frac{1}{4-3x} dx = ...
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1answer
230 views

Logarithmic terms

I've found a formula derivation in my international economics book, but I can't understand how it was derived. It says $ E P X = P^* IMP $ where E is the exchange rate, P is the level of national ...
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4answers
380 views

How to know if $\log_78 > \log_89$ without using a calculator?

I realize that I lack any numerical intuition for logarithms. For example, when comparing two logarithms like $\log_78$ and $\log_89$, I have to use the change-of-base formula and calculate the values ...
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4answers
288 views

Logarithm with quadratic solution

I have looked over this question several times, and I only understand the solution up to a point. Solve the equation for $x$: $$\ln x+\ln(x-1)=1 $$ First thing I do is apply the additive rule of ...
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1answer
108 views

If $ \log \log x =1$, then is it true that $e^e =x$?

I worked out this question, and I wanted to see if my understanding of the concepts involved is sound. Solve for $x$ $$\ln(\ln(x))=1$$ $$e^1=\ln(x)$$ $$e^e=x$$ Since any number raised to $1$ is ...
5
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1answer
986 views

How to figure of the Laplace transform for $\log x$?

I was looking at a table of common Laplace transforms of functions when I came across the transform for $\log x$. Apparently, the transform is as follows: $$\mathcal{L} \left\{ \log ...
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2answers
68 views

What is the math behind this transformation on exponents that are logarithms?

I understand that $$a^{\log_b(n)} = n^{\log_b(a)}.$$ What is the math behind this transformation that allows you to swap the $a$ and $n$?
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1answer
35 views

Specific range of numbers is given, trying to get another number within same range

I'm trying to calculate the width of an HTML element based on the window size. Here's what I have. These width values (first value) accurately match with the width the HTML element must be (second ...
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2answers
408 views

Determination of complex logarithm

Good day everyone. I was reading the more advanced lectures on complex analysis and encountered a lot of questions, concerning the determination of complex logarithm. As far I don't even understand ...
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4answers
23k views

How to figure out the log of a number without a calculator?

I have seen people look at log (several digit number) and rattle off the first couple of digits. I can get the value for small values (aka the popular or easy to know roots), but is there a formula. ...
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7answers
214 views

Is the equation $\log[A/(-B)]=\log[(-A)/B]$ valid?

A friend sent me these lines: $$\log[A / (-B)] = \log[(-A) / B]$$ $$\log(A) – \log(-B) = \log(-A) – \log(B)$$ $$\log(A) – [\log(-1) + \log(B)] = \log(-1) + \log(A) – \log(B)$$ $$\log(A) – \log(B) - ...
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3answers
157 views

Logarithmic non-integer fractional value

Would it be possible to show the breakdown of how $\log_4$ $32$ = $\frac{5}{2}?$ I have to come up w/ 11 more just like it & I'm not sure how you came up w/ the answer. Thank you!
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4answers
2k views

Inverse of the natural log function $y =\ln x$

Of course, it is a well known fact that the inverse of $y=\ln x$ (natural logarithm of x) is $e^x$. Assuming we haven't heard of the exponential function at all, how do we prove that the inverse of ...
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1answer
82 views

Does $\int \frac{1}{u} du = \ln|u| + C$ also work when $u$ is complex?

Does $\int \frac{1}{u} du = \ln|u| + C$ also work when $u$ is complex? I was taught this in calculus but I'm not sure if it generalizes to complex variables. Thank you!
4
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1answer
318 views

Super logarithmic inverse of tetration

What's the super logarithmic inverse of tetration for $\bf{^{2}{x}}$? Is it $slog^{x}_{2}$?
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0answers
63 views

Question about an asymptotic analysis proof in Ball Collision Decoding paper.

On page 21 of Daniel Bernstein's paper "Smaller decoding exponents: ball-collision decoding" he presents a proof that I have a few questions about. $P,Q,R,L$ and $W$ are all positive and close to ...
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1answer
116 views

Complex logarithm, my answer is wrong

I am trying to calculate $$\log(-1+i)$$ I have $$\log(-1+i) = \ln|(-1+i)| + i\operatorname{Arg}(-1+i)$$ $$ = \ln\sqrt2 + i3\pi/4$$ However when I checked that in matlab and wolfram alpha they have ...