Questions related to real and complex logarithms.

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2answers
44 views

Dividing logarithms without using a calculator

The problem I have is: $\log16+\log25-\log36\over{\log10-\log3}$ (log is base 10 here) I have the answer as 2 but no idea how to reach it.. I need to work this out without the use of a calculator ...
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0answers
24 views

Growth rate of bacteria involving logarithmic functions

I was trying to solve the following question but I keep getting the wrong answer, could anyone help me out and see why? A bacteria culture starts with 900 bacteria and grows at a rate proportional to ...
0
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1answer
30 views

Numbers of solutions of the equation $\log_3\frac {2x^2+3x+3}{5} = \frac {1}{\log_{2x^2+3x+9}9}$

Pretty straightforward question. When I solved it, I got two positive and two negative solutions, so that would make 4 in total. None get discarded as the arguments in the logarithm still stay ...
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0answers
23 views

Roots of serie $\sum_{i=1}^N \frac{a_i + b_i \log(c_i - x)}{c_i - x}$

I encountered the following serie during an optimization problem: $\sum_{i=1}^N \frac{a_i + b_i \log(c_i - x)}{c_i - x}$ I need to find the roots of that serie, but I cannot find the way to find a ...
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1answer
35 views

Do these functions have the same domain: $y=\ln (x^2)$ and $y=2\ln(x)$

I believe the domain of the first one has all the $x\in\mathbb{R}$ different from $0$ and the second is $x>0$, but aren't these two functions equal?
3
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0answers
52 views

How does $\ln(x)$ blow up at $0$ and $\infty$.

In general: How do I figure out how fast a function blows up at a certain point or infinity? How fast does $\ln x$ blow up at $0$? Does it blow up as fast as $1/x$, $1/x^2$, or maybe faster than any ...
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1answer
26 views

SAT2 Level 2 Book Answer Error

I am currently studying for my SAT2 Subject Test in Mathematics Level 2 and was check my answers to a practice test when I can across this (below) question. Problem: George invests $\$1000$ into ...
10
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2answers
76 views

Solve $\int_{0}^{1} \log(x)\log(1-x) dx$ without convolution

Maybe it's too much to ask for, but is there a way to solve $\int \limits_{0}^{1} \log(x)\log(1-x) dx$ without convolution? Note that $\log x =\log_e x$.
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0answers
17 views

I always have some doubts regarding the inequalities in cases where the function become Complex in the field for the real numbers

Consider this inequality $x + \log\left(x \right)> \log\left (x\right) - 2$ Does this inequality has $-1$ as its solution ? It will be very helpful for me.
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1answer
32 views

Discrete math, Showing a recursive equation as equivalent to a non recursive equation.

I'm having trouble with this: Show that this recursive function: $L(n) = \{0 : n = 1\ ,\ \lfloor(L(n/2))\rfloor +1 : n \gt 1\}$ is equivalent to this non-recursive equation: $L(n) = ...
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0answers
48 views

Logarithms $\log_8\boxed!+\log_87=\log_877$ [closed]

$$\begin{align} \log_211-\log_25&=\log_2\boxed!\\ \log_8\boxed!+\log_87&=\log_877\\ 5\log_62&=\log_6\boxed! \end{align}$$ I don't know where to start with this problem. The three ...
2
votes
1answer
43 views

Where to write the power with a logarithmic function?

This might be a simple question, but where would I write the power if I had a logarithmic function? Instinctively I would write it as $\log^y(x)$. But I'm not sure if this is correct. Should I be ...
0
votes
1answer
30 views

Real logarithm of a real matrix?

What is the real logarithm of \begin{equation} \begin{pmatrix} -1 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & -1 \end{pmatrix}? ...
1
vote
1answer
21 views

Prove that $ln(x)$ is concave by using the following definition for $ln(x)$

Using the following definition: $$\ln(x) = \int_1^x \frac{1}{t} dt$$ Show that ln is concave. So basically what I need to show is that for $x,y \in \mathbb{R}^+, x \neq y$ and for some $t \in (0,1): ...
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vote
2answers
47 views

How do you write bases on a computer? [closed]

I know you can write exponents for example, 2 to the power of 3 is usually written on a computer as 2^3 however, what if you are writing a base? Is their a different symbol? How would I write for ...
0
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3answers
38 views

Solve this equation: $\log_3(3-2\cdot3^{x+1})=2+2x$

Solve this equation: $\log_3(3-2\cdot3^{x+1})=2+2x$. I put $(2+2x)^3=3-2\cdot3^{x+1}$. But I don't know how to go on.
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3answers
38 views

Solve the equation: $x^{1/(1+\log x)}=10^3$

Solve the equation: $x^{1/(1+\log x)}=10^3$. I thought to take the logarithm on both sides but I couldn't find a solution.
2
votes
6answers
57 views

Solve this logarithmic equation: $2^{2-\ln x}+2^{2+\ln x}=8$

Solve this logarithmic equation: $2^{2-\ln x}+2^{2+\ln x}=8$. I thought to write $$\dfrac{2^2}{2^{\ln(x)}} + 2^2 \cdot 2^{\ln(x)} = 2^3 \implies \dfrac{2^2 + 2^2 \cdot ...
2
votes
2answers
36 views

Solve this equation $4^{\log_2(x)}-2^{\log_2(x)}=3^{\log_3(12)}$.

Solve this equation $4^{\log_2(x)}-2^{\log_2(x)}=3^{\log_3(12)}$ I thought to write $2^{\log_2(x)^2}-2^{\log_2(x)}=3^{\log_3(12)}$. Then is there a way to factorize $2^{\log_2(x)}$? I don't know how ...
0
votes
2answers
71 views

Integration of log(sinx)

In trying to integrate log(sinx) and I ended up looking for a solution and found the one at this link: http://www.meritnation.com/ask-answer/question/how-to-integrate-f-log-sin-x-dx/math/766517 ...
3
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5answers
2k views

How do I square a logarithm?

How do I square $\log_2(3)$. Does it become $2\log_2(3)$ ? If yes,Please explain.
2
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1answer
42 views

Maclaurin series for a function

Provided I have the function \begin{equation*} f(x)=(1+x)^{1/x}, \end{equation*} and I want to calculate a 3rd order Maclaurin series, how can that be done without taking direct derivatives (as ...
4
votes
2answers
59 views

integral of logarithm and rational function

i'm wondering how can i evaluate this integral using real methods: \begin{equation*} \int_{0}^{\infty}\frac{\log x}{1+x^{2}}dx. \end{equation*} I tried using mclaurin series of $\log x$ but really ...
4
votes
4answers
109 views

What does $d\log\left(\frac{y}{x}\right)$ mean mathematically?

I am used to seeing derivatives written as $$\frac{df}{dx}.$$ But my economics professor keeps using notation like $$ d\log\left(\frac{y}{x}\right)$$ and I have no idea what this means. What does ...
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0answers
260 views

Fermat's last theorem generalization [closed]

Conjecture: Let $g$ is a positive algebraic number greater than two, then the equation ($x^g+y^g=z^g$) doesn't have any solution, where ($x, y$ and $z$) are three distinct positive coprime integers ...
2
votes
2answers
67 views

Find the value of $x$ that satisfies the equation $\log_{10} \left(\frac{x^{\frac{1}{x}}}{x^{\frac{1}{x+1}}}\right) = 1/5050$ .

I tried it many times and it went bit of lengthy , i reached until \begin{equation*} \log_{10}(x^{1/(x^2+x)}) \end{equation*} then i multiplied $2$ both numerator and denominator and then it is ...
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0answers
32 views

Complex Line integral of 1/z over the principle branch cut

I would appreciate it if someone checked my work to ensure that it's consistent. Compute the integral $\int_{C}\frac 1 z {dz}$ by obtaining an appropriate branch of the logarithm. There's an ...
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1answer
28 views

How do I prove that ab+5(a-b)=1

If $\log_{12}18=a$ and $\log_{24}54=b$ then how do I prove $ab+ 5(a-b)=1$? I figured that out it's $\log_ab$ and $\log_{2a}3b$ but how do I solve it?
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2answers
34 views

How do i get from $x^{(\log(x))}=10000 $ to $\log(x)^2=\log(10000)$

I'm looking at the solution for a math problem I'm trying to solve and can't comprehend the following step: From: $$ x^{\log_{10}(x)}=10000 $$ To: $$ {\log_{10}(x)}^2=\log_{10}(10000) $$ Is there a ...
0
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2answers
23 views

Right angled triangle log

If $a,b$ and $c$($c$ is the hypotenuse) are sides of a right triangle then prove $$(\log_{c+b}a)+(\log_{c-b}a)=2(\log_{c+b} a )\cdot(\log_{c-b}a)$$ The bases are different so can't quite figure out ...
2
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4answers
44 views

Domain of the function $f(x) = \sqrt{\frac{3^x-4^x}{x^2-4x-4}}$ will be?

I tried solving this question by $1.$ $-1$ and $4$ will not be in domain because denominator can not be zero . $2.$ Either both denominator and numerator will be positive or negative so that whole ...
4
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1answer
39 views

Way to calculate exponent in congruent equation

I want to solve $$ 5^{x} \equiv 21 \pmod {23} $$ Is there a way to get the $x$ without trial & error?
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0answers
13 views

How do we get $S(m) = S(m/2) + \lg m$ from $T(n) = T(\sqrt{n}) + \lg\lg n$?

I am confused about example we got today in class. Here is a recurrence and I am not sure how we got $S(m)=S(m/2)+(\lg m)$ $$T(n)=T(\sqrt{n}) + (\lg\lg n) $$ Let $$m =\lg n$$ $$S(m)=S(m/2)+(\lg m) ...
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0answers
12 views

Logarithm with logarithm table

Out of curiosity, I wanted to use a log table to find the the logarithm of 347,5. (I always used a calculator) The first three digits (347) give me 5403 for the mantissa and for the fourth figure (5) ...
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1answer
26 views

A problem with logarithms

If $\log(a+b+c)=\log(a) + \log(b) + \log(c)$, prove that $$\log\left(\frac{2a}{1-a^2} +\frac{2b}{1-b^2} +\frac{2c}{1-c^2}\right) = \log\left(\frac{2a}{1-a^2}\right)+ ...
1
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1answer
20 views

Finding out the logarithmic function for the situation below

The situation reads as follows: There are 3000 barbs in a pond and every year 20% of the barbs die and then 1000 new barbs come to the pond. A logarithmic function needs to be plotted to graph ...
0
votes
3answers
45 views

How does $\log_2(A)-\log_2(B)+\log_2(c)$ not equal $\log_2(\frac{Bc}{A})$

$\log_2(A)-\log_2(B)+\log_2(c)$ How does this equal $\log_2(\frac{Ac}{B})$? Does it not follow from the order of operations that it would be addition first then subtraction? I'm having hard time ...
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4answers
55 views

A rational number which is 50 times its own logarithm to the base 10 is?

This question is from Advanced problems in mathematics for jee . I got it as a challenging question. I tried it in this way 50 log x base 10 = x But there seemed no solution for it as per my ...
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4answers
72 views

How to prove that $\frac{\ln 12}{\ln 18}$ is irrational witout using the change of base rule? [closed]

I have to show that $\frac{\ln 12}{\ln 18}$ is irrational by using change of base rule. At the beginning I have proved that $\ln r$ is irrational for any rational $r$, $r\ne 1$. Then using this we ...
2
votes
4answers
258 views

Explain this inequality, related to logarithms

I am trying to understand a proof of Stirling's formula. One part of the proof states that, 'Since the log function is increasing on the interval $(0,\infty)$, we get $$\int_{n-1}^{n} \log(x) dx ...
5
votes
5answers
103 views

Is it possible to prove this? $\ln(\frac{x}{x-1}) < \frac{100}{x} $ for $ x > 1$

$-\ln(1-(\frac{1}{x})) < \frac{100}{x} $ for $ x > 1$ is what I want to prove. I pulled a negative sign out and I got $\ln(\frac{x}{(x-1)}) < \frac{100}{x} $ for $ x > 1$. How do I ...
0
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2answers
34 views

Meaning of exponent in logarithm?

I have this particular difficulty : $$\log_b^a(c)=x$$ I know it is different from power of base $\log_{b^a}(c)=x$, but what does it actually mean? The actual question that i got in paper was Find ...
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1answer
43 views

Simplifying logarithms into a single log using log Laws [closed]

I need some help on how to solve this question. $$1.5(\log_bx+2\log_by^4)-0.5(\log_b\sqrt x+\log_by^{1/3})$$ Shall appreciate some help on this. My Work: if possible, could u tell me if my ...
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2answers
31 views

Simple math Question concerning the natural logarithm of Complex Number

There is this simple exercise, in which the complex number is given in polar form as z= mod=|10|,arg=322.75 degrees and i must find the ln of it. So to do that i must first convert the complex number ...
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1answer
36 views

Unable to solve logarithm question

Given $$\dfrac{a(b+c-a)}{\log a}=\dfrac{b(c+a-b)}{\log b}=\dfrac{c(a+b-c)}{\log c}$$ To prove: $$a^bb^a=b^cc^b=c^aa^c$$ What i tried is $$\log (a^z)=a(b+c-a)$$ and similarly for other two. I am ...
2
votes
1answer
43 views

Is this $\lim \ln(f(x))=\ln(\lim(f(x))$ valid?

Is this mathematically legit? $$\lim_{x\to\infty}\ln(f(x))=\ln(\lim_{x\to\infty}(f(x))$$
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3answers
32 views

Logarithm formula proof

Prove: $$x^{\log(y)}=y^{\log(x)}$$ I have been trying this for the past 1 hour, still cant prove it. I started with $$\log_b(y)=m$$ $$\log_b(x)=n$$ To show: $$x^m = y^n$$ How do i proceed? :
2
votes
1answer
21 views

Find curve that fits (min, mean, max) to (0, 0.5, 1) [closed]

I'm trying to use the fact that $log(1) = 0$ and $log(\sqrt{e}) = 0.5$ and $log(e) = 1$ to write a map from a set of data points to a value between $0$ and $1$ such that: $f(min) = 0$, $f(mean) = ...
0
votes
2answers
37 views

Showing that for continuous logarithms $g_1, g_2$ of a function on a connected set, the difference $g_1 −g_2$ is a constant

If $S$ is connected, $\ f$ is continuous and has continuous logarithms $g_1$ and $g_2$ on $S$, and continuous arguments $\theta_1$ and $\theta_2$, then $g_1 −g_2$ and $\theta_1-\theta_2$ are ...
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1answer
25 views

Solving a logarithmic equation with variables on each side

Okay, so while doing a problem for my calculus class I was required to graph two functions in order to see where they intersect, as according to my teacher there is no way to solve it analytically. ...