Questions related to real and complex logarithms.

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1answer
319 views

simple calculation using logs

Suppose we are comparing implementations of insertion sort and merge sort on the same machine. For inputs of size $n\in\mathbb{N}$, insertion sort runs in $8n^2$ steps, while merge sort runs in ...
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3answers
790 views

Logarithms explained simply

Sorry for the trivial question. If I have the expression $\log(5)$, and the base is $10$, what operation is being performed on the number $5$, in words? For example, I know that exponents work (say ...
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2answers
198 views

How do you solve this simple logarithm problem?

I'm comparing efficiencies for the famous fake-coin algorithms. Specifically, I'm looking at a two-pile approach and a three-pile approach for a solution. I have found that, like a binary search, ...
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3answers
160 views

Evaluate $\int_0^1 {\ln(1+x)\over x}\,dx$.

How would one evaluate $\int_0^1 {\ln(1+x)\over x}\,dx$? I'd like to do this without approximations. Not quite sure where to start. What really bothers me is that I came across this while reviewing ...
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1answer
163 views

How to calculate $\int_{|z|=r}\ln(1-z)\,dz$ in dependence of $r\neq1$?

With the integration I mean one counter-clockwise turn around the origin, i.e. $$\int_{\phi=0}^{2\pi}\ln(1-re^{i\phi})ire^{i\phi}d\phi$$ For $r<1$, this is simply a contour integration on a ...
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3answers
426 views

How can I solve $8n^2 = 64n\,\log_2(n)$

I currently try to analyze the runtime behaviour of several algorithms. However, I want to know for which integral values $n$ the first algorithm is better ($f(n)$ is smaller) and for which the second ...
2
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1answer
147 views

Solving logarithmic equations of of the form $\ln(xa)= b\ln(c-x)$

Given: $$\ln(xa)= b\ln(c-x)$$ I am unsure of how to manipulate the values within the natural logs to solve for x while the factor b remains. I can safely move in circles by applying the definition ...
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4answers
262 views

How to find the limit $\lim \limits_ {x \to+\infty} \left [ \frac{4 \ln(x+1)}{x}\right]$

Solve $\space \begin{align*} \lim_ {x \to+\infty} \left [ \frac{4 \ln(x+1)}{x}\right] \end{align*}$. I did this way: $$\begin{align*} \lim_ {x \to+\infty} \left [ \frac{4 \ln(x+1)}{x}\right] ...
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1answer
85 views

Equation model for project effort.

It is my first time here, so I hope I'm keeping on topic. I wanted to find an equation where I could use the variables which affect the amount of work, in a way that feeling it the variables, I'd ...
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6answers
486 views

Proof that $\int_1^x \frac{1}{t} dt$ is $\ln(x)$

A logarithm of base b for x is defined as the number u such that $b^u=x$. Thus, the logarithm with base $e$ gives us a $u$ such that $e^u=b$. In the presentations that I have come across, the author ...
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2answers
519 views

Why are $\log$ and $\ln$ being used interchangeably?

A definition for complex logarithm that I am looking at in a book is as follows - $\log z = \ln r + i(\theta + 2n\pi)$ Why is it $\log z = \ldots$ and not $\ln z = \ldots$? Surely the base of the ...
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4answers
236 views

Solving the exponential equation: $3 \cdot 2^{2x+2} - 35 \cdot 6^x + 2 \cdot 9^{x+1} = 0$

I have this exponential equation that I don't know how to solve: $3 \cdot 2^{2x+2} - 35 \cdot 6^x + 2 \cdot 9^{x+1} = 0$ with $x \in \mathbb{R}$ I tried to factor out a term, but it does not help. ...
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3answers
449 views

Are the logarithms in number theory natural?

I find the frequent emergence of logarithms and even nested logarithms in number theory, especially the prime number counting business, somewhat unsettling. What is the reason for them? Has it maybe ...
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1answer
49 views

Formula to calculate when you will have a certain amount of money in your bank account

What is the formula to calculate when I will have a million dollars in my bank account? An example is that I have $\$6,000$ in my account and have $6\%$ interest rate on that. How long will it take ...
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2answers
255 views

Can all logarithm problems be solved algebraically?

Trying to solve $\log_2(x-1)=\log_3(x+1)$ and can't seem to get it algebraically. Tried changing bases, moving things around, but can't seem to crack it.
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1answer
307 views

The definition of the logarithm.

One usually gets several definitions of the logarithm along his studies. You might be first introduced to the exponential and then told that the logarithm is its inverse. You might be given $$\log ...
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3answers
10k views

From natural log to log base 10

The constraints of this question is related to a programming problem, but I must get the math right in order for it to be applied to code. The actual problem is I need a function that evaluates to log ...
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2answers
438 views

continuum between linear and logarithmic

A friend and I are working on a heatmap representing some population numbers. Initially we used a linear color scale by default. Then, because the numbers covered a wide range, I tried using a log ...
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2answers
203 views

Prove $ n+1<\frac{\log 4}{\log3}+\frac{\log 44}{\log33}+\frac{\log4444}{\log3333}+\cdots+\frac{\log 444\ldots444}{\log333\ldots333} <n+2 $

Prove that $$ n+1<\frac{\log 4}{\log3}+\frac{\log 44}{\log33}+\frac{\log4444}{\log3333}+\frac{\log 44444444}{\log33333333}+\cdots+\frac{\log 444\ldots444}{\log333\ldots333} <n+2$$ where last ...
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1answer
92 views

Does the logistic function really uniquely satisfy this?

It is said that the logistic function (denoted $y(u)$ below) is derived from the relation: $$\frac{dy}{du}=y(u)(1-y(u))$$ Does $y(u)=\frac{1}{1+e^{-u}}$ really uniquely satisfy this? I don't see ...
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1answer
431 views

closed form solution for summation of $\log(i)$

Is there a way to find a closed form solution for: (Note that base is $2$) $\displaystyle\sum_{i=1}^n\log_2(i)$ thanks for any help Can't find a formula for this
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2answers
267 views

Graph of a Log Function

I am curious as to why Wolfram|Alpha is graphing a logarithm the way that it is. I was always taught that a graph of a basic logarithm function $\log{x}$ should look like this: However, ...
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1answer
303 views

Reverse of a log-based f(x)?

I am attempting to create a function, f(y) that can reverse my existing function, f(x). ...
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1answer
160 views

Are all logarithms multiple of each other?

I was doing a time complexity problem, and the solution mentioned that there is a single class for logs. Ie. we can write $\log_a (x) = \Theta(\log_b(x))$ where $a$ is not equal to $b$. This can be ...
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1answer
107 views

Solutions for this logarithmic equation.

For which values of $k$ does the equation $\log_a(kx+3)+\log_a(x+1)=\log_a(2x+1)$ have one or more solutions in $x$? The logarithmic functions must have the restriction that the argument is ...
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3answers
323 views

Logarithms of the form $x=e^y$

I have the following math problem: The number of people in a town of 10,000 who have heard a rumor started by a small group of people is given by the following function: $N(t) = ...
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3answers
1k views

Solve 10 base logarithms

I'm a n00b in math and I wanted to know how should I solve ten base logarithms. e. g.: Log 40 to base 10 Thanks in advance.
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3answers
206 views

Solving $\int\frac{\ln(1+e^x)}{e^x} \space dx$

I'm trying to solve this integral. $$\int\frac{\ln(1+e^x)}{e^x} \space dx$$ I try to solve it using partial integration twice, but then I get to this point (where $t = e^x$ and $dx = \frac{1}{t} ...
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3answers
111 views

Comparing numbers in form $x^y$

Let's consider two numbers in form $x_1^{y_1}$ and $x_2^{y_2}$ How can we compare those two numbers without evaluating them ? Can we use logarithms to check it ? If yes - how ? Thanks in advance. ...
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3answers
273 views

Does $\log(ab)^n$ equal $(\log(a)+\log(b))^n$ or $n\log(a)+n\log(b)$?

I think this might be a case of slight ambiguity in notation, but here goes: On a test question, I was required to expand the expression $\log (ab)^n$. Since the logarithm is a function, I reasoned ...
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3answers
941 views

arithmetic progression involving logarithm

$\log_2 X$, $\log_2 (X+9)$ and $\log_2(X+45)$ are 3 consecutive terms of an arithmetic progression; find $\qquad$(i) the value of X; $\qquad$(ii) the first term and the common difference; and ...
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2answers
106 views

Limit with prime sequence and inverse logintegral

I found formula below$$\lim_{n\to\infty}\frac{\operatorname{li^{-1}}(n)}{p_n}=1$$ where $\operatorname{li^{-1}}(n)$ is inverse logintegral function and $p_n$ is prime number sequence. Can anyone ...
5
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2answers
217 views

Why isn't $\frac{\mathrm{d} }{\mathrm{d} x} \ln(x)$ specified as $\frac{1}{x},x>0$?

As I understand, $\begin{eqnarray} \frac{\mathrm{d}}{\mathrm{d}x}\ln(x)\end{eqnarray} $ is generally specified as $\begin{eqnarray} \frac{1}{x} \end{eqnarray}$. Would it be more appropriate to state ...
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4answers
566 views

How to evaluate $\int_{1}^{2}\frac{dx}{1+x+\ln x}$?

Can you help me find the value of the integral $$\int_{1}^{2}\frac{dx}{1+x+\ln x}$$ Thank you
5
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1answer
380 views

Inverse function of $\operatorname{li}(x)$ over $x>\mu$?

How can I get the inverse function of $\operatorname{li}(x)$ over $x>\mu$? Where $$\operatorname{li}(x)=\int_{0}^{x}\frac{ds}{\ln(s)}$$ is the so-called logarithmic integral, and ...
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1answer
79 views

How to get “N” from $k=\log_2(N)$

I know it's a easy question but unfortunately I forgot some school stuff: I have $k=\log_2(N)$ and want to know $N$. Is it $N=2^k$ while using $2$ as base? Short comments are welcome :)
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5answers
753 views

How to solve for negative numbers in logarithmic equations

I am trying to solve the equation $$z^n = 1.$$ Taking $\log$ on both sides I get $n\log(z) = \log(1) = 0$. $\implies$ $n = 0$ or $\log(z) = 0$ $\implies$ $n = 0$ or $z = 1$. But I clearly missed ...
2
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1answer
158 views

Find the limit of a sequence defined as solution to equation

We can easily prove that the equation of variable $x$ $$(E_{n}): \frac{x(\ln x)^n}{1+x}=\frac{e}{2(e+1)}$$ has a unique solution $u_{n}$ in $[1,e]$ for all integers $n$ greater than $1$. Let's call it ...
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2answers
97 views

Trying to figure out how an approximation of a logarithmic equation works

The physics books I'm reading gives $$\triangle\tau=\frac{2}{c}\left(1-\frac{2m}{r_{1}}\right)^{1/2}\left(r_{1}-r_{2}+2m\ln\frac{r_{1}-2m}{r_{2}-2m}\right).$$ We are then told $2m/r$ is small for ...
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3answers
340 views

Inequality for logarithms

I conjecture the following inequality is true $$\ln x \le (x - 1)\ln\frac{x}{x-1}$$ for all $x > 1$, but I cannot give a proof. I will appreciate if someone can provide one.
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1answer
188 views

How to solve a literal equation

How do I solve $2^{x-1}=3^{x+a}$? I cannot solve it and have spent an hour on it trying many different ways. Please help me! Thank you!
3
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1answer
302 views

predicting runtime of $\mathcal{O}(n \log(n))$ algorithm, one “input size to runtime” pair is given

I'm given the runtimes for input size $n=100$ of some polynomial-time (big-Oh) algorithms and an $\mathcal{O}(n \log(n))$ one. I want to calculate the runtimes for: $200$, $1000$ and $10000$. For the ...
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2answers
154 views

How to solve $5 - \log_2 (x - 3) = \log_2(x+1)$

Sorry I have to ask such a simple question, my brain is fried after today. After substituting with a system of equation, I end up with this "simple" logarithmic problem. $$5 - \log_2 (x - 3) = ...
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1answer
397 views

How to find logarithms of negative numbers?

Logarithms of negative numbers must be complex. But how do you find $\ln{(-2)}$ expressed in something like $x \cdot i$ where $x \in \mathbb{R}$?
3
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2answers
123 views

Simple logarithmic equation

If $y=Ae^{-kt}$ and $y=19.6$ when $t=2$, and $y=19.02$ when $t=5$, find the value of the constants $A$ and $k$. Give your answers correct to $2$ decimal places. I have spent a while (an hour+) on ...
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1answer
192 views

From half to double, linear to logarithmic scale.

I am making a game where you want a skill value to modify some in game values. With a scale that goes from half to double. 50% to 200%. If I'd do it linear 125% will be the centre but I want the ...
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3answers
144 views

Squeeze an integral

Would you have any idea about this problem ? Prove that for all nonnegative integers $n$, the following inequalities hold: $$\frac{e^2}{n+3}\leq \int_{1}^{e} x (\ln x)^n \,dx \leq ...
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1answer
277 views

Logarithmic differentiation for this function

What is the value of $f'(x)$ at $c$, when $f(x) = \log_x c = e$? (I understand the answer could be $1/e$) but am unable to substantiate the reasoning. Can someone please help me take the ...
2
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1answer
336 views

Find solution to system with logarithms

I have two equations: \begin{align*} 3 \ln x + \ln y &= 3 \\ 4 \ln x - 6 \ln y &= -7 \\ \end{align*} Do I just proceed as I have learned with adding equations resulting in: \begin{align*} ...
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1answer
74 views

Basic logarithm simplification

From $\displaystyle \frac{\log_n b}{\log_n a}$, how do we get $\log_a b$ using algebra? I haven't been able to do it for about an hour now; I would love some help! Thanks!