Questions related to real and complex logarithms.

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13
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8answers
864 views

Why does the logarithm require a special notation?

Since the logarithm is the reversed exponentiation, why does it need a distinct notation for it? Why can't we just ask: $$2^x=8$$ Instead of: $$\log_2 8=x$$
0
votes
2answers
79 views

solving in x involving both exponential and logarithmic function

Is it possible to solve a function with both exponential and logarithm such as $a x^2−b.\log(x)= c$ in closed form; where $a,b,c$ are constants and $a>0$ and $b>0$?
1
vote
2answers
203 views

Finding a logarithmic function from a graph

Here's the graph. When I use the points $(-1,1)$ or $(-3,2)$ to use in the equation $a\log(-x-1)+k$, I can't find a finite value for k. Any ideas?
1
vote
2answers
1k views

Graph $f(x)=\ln x+2$

Graph $f(x)=\ln x+2$ And find all intercepts and asymptotes. I know exactly how the graph looks and I have a sketch in front of me. Now, for the intercepts, $x$-int=$-1$ and $y$-int$=-\infty+2$ ...
2
votes
2answers
158 views

Graph $f(x)=e^x$

Graph $f(x)=e^x$ I have no idea how to graph this. I looked on wolframalpha and it is just a curve. But how would I come up with this curve without the use of other resources (i.e. on a test).
4
votes
4answers
2k views

Find the domain of $f(x)=\ln(3x+2)$

Find the domain of $f(x)=\ln(3x+2)$ I can find domain, but is it the same for a $\log$ function? And also, do I have to rid the equation of the $\ln$ before I can find the domain? I'm really ...
2
votes
2answers
1k views

Expand $\ln\left[\frac{(4x^5-x-1)\sqrt{x-7}}{(x^2+1)^3}\right]$.

Expand this expression to the greatest possible terms with the lowest possible exponents. $\ln\left[\dfrac{(4x^5-x-1)\sqrt{x-7}}{(x^2+1)^3}\right]$ There are two ways at which I approached this ...
7
votes
4answers
191 views

If $x, \log_{10}(x), \log_{10}\log_{10}(x)$ are in arithmetic progression, find the range of $x$.

If $x, \log_{10}(x) , \log_{10}\log_{10}(x)$ are in arithmetic progression, find the range of $x.$ (a) $0 < x < 1$ (b) $1 < x < 10$ (c) $10 < x < 100$ (d) $100 ...
0
votes
1answer
410 views

Why does $\ln|\cot x|=\ln|\cos x|-\ln|\sin x|$ hold?

I am learning trigonometry. I can solve simple trigonometric equations. But its integration with log always confuses me. I am thinking about this sums since last two hours but can't find the ...
0
votes
1answer
149 views

Is there a relationship between $\ln(x\pm y)$ and $\ln(x)\pm\ln(y)$?

I am dealing with points on a 2d space $(x, y)$ where $x$ and $y$ are always positive integers. In an algorithm, I have pre-computed $\log_2(x)$ and $\log_2(y)$ for given points of interest. I now ...
4
votes
3answers
182 views

How to prove this ln inequality?

I have the following inequality, which (supposedly) holds for every $x\in\mathbb{R}$: $$ 1+x\ln\left(x+\sqrt{1+x^{2}}\right)\geq\sqrt{1+x^{2}} $$ I've been struggling to find some known inequalities ...
1
vote
2answers
85 views

Multiple regression with model $Y = (1 + c_1X_1)(1 + c_2X_2)\ldots(1 + c_nX_n)$

I'm currently working with data contained in $Y, X_1, X_2, \ldots, X_n$ and wish to fit it to the model: $Y = (1 + c_1X_1)(1 + c_2X_2)\ldots(1 + c_nX_n)$ where the $c_i$ are coefficients to be ...
5
votes
1answer
1k views

$\log_7 n$ is either an integer or an irrational number

Show that $\log_7 n$ is either an integer or an irrational number where n is a positive number. I assumed that it is rational and tried to get a contradiction for $\log_7 n = a/b$, where b does ...
2
votes
1answer
7k views

Why the number e(=2.71828) was chosen as the natural base for logarithm functions? [duplicate]

Possible Duplicate: What's so “natural” about the base of natural logarithms? Why the number e(=2.71828) was chosen as the natural base ...
1
vote
3answers
508 views

Show $(\log n)^{ (\log n) } = 2^{(\log n)(\log (\log n))}$

I am having difficulty understanding how this follows. $$(\log n)^{ (\log n) } = 2^{(\log n)(\log (\log n))} = n^{\log \log n}$$ Which logarithmic identities are used to go through each equality? ...
4
votes
2answers
1k views

Intuition for iterated function for log log log n.

Intuitively, $\log n$ (base 2) is the number of times you have to divide $n$ by 2 before reaching a number around 2. (Waving our hands a little to gloss over floor vs ceiling and $\pm$ 1 errors.) ...
5
votes
1answer
271 views

Explain this code to compute $\log(1+x)$

It's well known that you need to take care when writing a function to compute $\log(1+x)$ when $x$ is small. Because of floating point roundoff, $1+x$ may have less precision than $x$, which can ...
3
votes
3answers
102 views

Determining and (dis)proving if $ \sum_{n = 1}^{\infty} (-1)^{n + 1} \left( 1 - n \log \left( \frac{n + 1}{n} \right) \right) $ converges

I am trying to determine if $ \sum_{n = 1}^{\infty} (-1)^{n + 1} \left( 1 - n \log \left( \frac{n + 1}{n} \right) \right) $ converges using an alternating series test. The test in question requires me ...
1
vote
1answer
115 views

Logarithms and prime factors

Let a-f be integers g.t. 2 with $a < b < c < d < e < f$. Let $$\ln def - \ln a b c = \alpha.$$ Let $\{p_i\}$ be the set of prime factors (with repetitions) in a,b,c. Let $\{q_i\}$ be ...
0
votes
2answers
196 views

Find the limit of $f(x)$ involving a sum of logarithms.

I need to find $\lim_{x\rightarrow0}f(x)$ for the following function: $f:(0,+\infty)$ $f(x)=[1+\ln(1+x)+\ln(1+2x)+\dots+\ln(1+nx)]^\frac{1}{x}$ I tried writing the logarithms as products: ...
3
votes
4answers
253 views

If $\log 0.318 = x$ and $\log 0.317 = y$, can $\log 0.319$ be expressed in terms of $x$ and $y$?

If $\log 0.318 = x$ and $\log 0.317 = y$, can $\log 0.319$ be expressed in terms of $x$ and $y$ ? Is there any way or we have to find $\log 0.319$ using log tables only? I'm not getting any ...
3
votes
1answer
195 views

Log as the inverse of Exp in the complex plane

It is standard practice to define on $\mathbb{C}$, $$\operatorname{Log}(z) = \log(|z|) + i \operatorname{Arg}(z).$$ When composed with $\exp$, we get $\operatorname{Log} \circ \exp (z) = z$, the ...
1
vote
1answer
60 views

Condensing logarithms

Simplify: $2\log_{10}\sqrt{x}+3\log_{10}x^{\frac{1}{3}}$ I got to this: $2\log_{10}x^{\frac{1}{2}}+3\log_{10}x^{\frac{1}{3}}$. Now, usually you bring the exponent the the front and that ...
7
votes
2answers
596 views

Motivation for definition of logarithm in Feynman's Lectures on Physics

I'm not sure if the title is descriptive enough; feel free to change it if you come up with something better. I've been reading through Feynman's Lectures on Physics. In the first volume, he ...
0
votes
3answers
416 views

Different log bases

I have done many $\log$ problems but I've never learned something such as $\log_ax-\log_by$. I know that to condense a logarithm you must have the same base: ...
3
votes
2answers
378 views

Why is $\log(b,n) = \lfloor \log_b(n) \rfloor$ primitive recursive?

I read in an introduction to primitive recursive function and Wikipedia that $$\log(b,n) = \lfloor \log_b(n) \rfloor$$ is primitive recursive. But how can that be? Is there any easy proof (and ...
2
votes
4answers
252 views

Is the function $y=\ln x^2$ the same as $y=2\ln |x|$?

Suppose I have a function $$y=\ln x^2$$ Then is this function the same as $$y=2\ln |x|?$$
1
vote
2answers
806 views

How to solve this logarithm system?

I am new to logarithms and I am having trouble with this logarithm system. \begin{align*} \log_9(x) + \log_y(8) & = 2, \\ \log_x(9) + \log_8(y) & = 8/3. \end{align*} A step-by-step ...
4
votes
2answers
146 views

Solve $ \left( \log_3 x \right)^2 + \log_3 (x^2) + 1 = 0$

I'm new to logarithms and I am having trouble solving this equation $$ \left( \log_3 x \right)^2 + \log_3 (x^2) + 1 = 0.$$ How would I solve this? A step-by-step response would be appreciated. ...
1
vote
2answers
152 views

Prove the statement : $\log(k + 1) -\log k>\frac{ 3}{10k}$

Prove the statement : $\log(k + 1) - \log k > \frac{3}{10k}$ Approach : $$\log(k+1)-\log{k} > \frac{3}{10k}$$ Clearly, $k\in\mathbb{Z}^{+}$ ...
4
votes
3answers
869 views

How popular and used were logarithm tables?

I've heard that, for a time, logarithm tables "sold more than the Bible". Can someone produce some reliable documentation about how prevalent they were ? Would a common shopkeep have one ? Would a ...
1
vote
1answer
162 views

What does it mean to be: “at least logarithmic”?

I am going through some CS basics and the documents says in places that: We need the run time to be at least logarithmic. What does that mean? By def, log means: The logarithm of a number ...
1
vote
3answers
383 views

$X = \log_{12} 18$ and $Y= \log_{24} 54$. Find $XY + 5(X - Y)$

$X = \log_{12} 18$ and $Y= \log_{24} 54$. Find $XY + 5(X - Y)$ I changed the bases to 10, then performed manual addition/multiplication but it didn't yield me any result except for long terms. Please ...
1
vote
1answer
105 views

Prove that $\log \log y = \mathcal{o}(\log y) + \mathcal{O}(1)$.

I just wanted some help to prove that $$\log_2 \log_2 y = \mathcal{o}(\log_2 y) + \mathcal{O}(1),$$ when $y = f(n) \in \mathcal{O}(n)$ and $y > 4$. Thanks!
1
vote
6answers
233 views

How to find $\lim\limits_{n\rightarrow \infty}\frac{(\log n)^p}{n}$

How to solve $$\lim_{n\rightarrow \infty}\frac{(\log n)^p}{n}$$
0
votes
1answer
135 views

first 1 in a bitmask using log2

I am trying to get the last 1 in a bitmask. More mathematically speaking, I have a number k, that can be written in its binary form as a sequence of 1 and 0. I want the "weight" or "index" of the last ...
1
vote
4answers
207 views

Examples of logs with other bases than 10

From a teaching perspective, sometimes it can be difficult to explain how logarithms work in Mathematics. I came to the point where I tried to explain binary and hexadecimal to someone who did not ...
1
vote
5answers
183 views

How can I calculate $\lim_{x \to 0} \log(\cos(x))/\log(\cos(3x))$ without l'Hopital?

How can I calculate the following limit without using, as Wolfram Alpha does, without using l'Hôpital? $$ \lim_{x\to 0}\frac{\log\cos x}{\log\cos 3x} $$
2
votes
4answers
101 views

Show these simple inequalities

Show that $(\log(1+x))^2\le x$ and that $(\log(1+x))^2\le x^2$ for all $x\ge0$. Both of these inequalities seem to be true, judging from plotting these functions with a grapher. Can you help me ...
0
votes
1answer
97 views

Is $\log_{5}{-3} = \frac{\log(3)+\pi i}{\log(5)}$?

Why does my calculator return false when I input $\log_{5}{-3} = \frac{\log(3)+\pi i}{\log(5)}$ but W|A returns true? I'm thinking my calculator is wrong because I know that $\displaystyle ...
1
vote
3answers
85 views

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
votes
4answers
113 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 ...
1
vote
2answers
392 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 ...
0
votes
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
2
votes
4answers
215 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
votes
1answer
325 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 ...
5
votes
3answers
186 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 : ...
2
votes
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 ...
18
votes
2answers
303 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 ...
5
votes
1answer
162 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 ...