Questions related to real and complex logarithms.

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2
votes
4answers
44 views

Gradient of a curve $y=\ln \sqrt{x+y}$

Find the gradient of the curve $y=\ln \sqrt{x+y}$ at the point when its y-coordinate is 1. My attempt, I differentiated and I got $\frac{dy}{dx}=\frac{1}{2x+2y-1}$. But I've problem in finding the ...
0
votes
2answers
28 views

Proving logarithm question

Prove: $$\log_a (bc)\times \log_b (ac)\times \log_c (ba)=2+\log_a (bc)+ \log_b (ac)+ \log_c (ba)$$ I took LHS and applied base change formula. I changed base to $`\text{abc'}$ Let $abc=\mu$ ...
0
votes
1answer
15 views

Interval of the solutions to $\log_{1/2}\log_2(\frac{1+2x}{1+x})>0$ is?

I consistently get $x>-1$ but that doesn't fit the possible solutions I've got. First step I do is state that $\log_2(\frac{1+2x}{1+x})<1$ Then express the $1$ as $\log_22$ and so on. What ...
2
votes
7answers
63 views

Another combined limit

I've tried to get rid of those logarithms, but still, no result has came. $$\lim_{x\to 0 x \gt 0} \frac{\ln(x+ \sqrt{x^2+1})}{\ln{(\cos{x})}}$$ Please help
-2
votes
2answers
40 views

Integral with logarithm is positive

Given the following integral: $$I(f) = \int_\mathbb{R} f(x) \log \left(f(x) \sqrt{2\pi} e^{\frac{x^2}{2}}\right) dx,$$ where we assume $\int_{\mathbb{R}} f(x)\, dx =1$ and $f\geq 0$ a.e. Assume for ...
1
vote
1answer
38 views

How to solve for x in $2^{2x^2}+2^{x^2 + 2x + 2} =2^{5+4x}$

This is the question: $$\large{2^{2x^2}+2^{x^2 + 2x + 2} =2^{5+4x}}$$ What I did was put $~\large{2^{x^{2}}=t}$ From this, I got, roots of the quadratic: $$\large{-2^{x+1}\pm~\left( ...
3
votes
4answers
212 views

Solving equations with exponentials and a non-exponential term.

I know how to solve exponential equations. Just use logarithms, e.g., $$ 2^x-3=0 \\ 2^x=3 \\ x=log_23 \\ $$ But on a recent math test I found an equation of the form: $$ 2^{n-3}=\frac {20}{n} $$ ...
1
vote
0answers
27 views

Proof $\log(cn)$ is in $\Theta(\log(n))$

How can I prove that $\log(cn)$ is in $\Theta(\log(n))$, where $c$ is a constant? I tried to prove $c_1\log(n) \le \log(cn) \le c_2\log(n)$, where $c_1$ and $c_2$ are also constants, but I'm having ...
3
votes
4answers
54 views

Solve the equation $\log_{2} x \log_{3} x = \log_{4} x$

Question: Solve the equations a) $$\log_{2} x + \log_{3} x = \log_{4} x$$ b) $$\log_{2} x \log_{3} x = \log_{4} x$$ Attempted solution: The general idea I have been working on is to make them ...
3
votes
3answers
124 views

antiderivative of $\frac{1}{z(z-1)}$, complex logarithm

I have the domain $\mathbb{C} \backslash [0,1]$ and want to show that $$\int_\gamma \frac{1}{z(z-1)}dz = 0$$ for all closed curves $\gamma$. I want to accomplish this by explicitly finding an ...
0
votes
2answers
29 views

need approach to solve given logarithm expression

I was going through algorithm on sorting and encountered a logarithm problem which need to be solved. Question Statement is: For inputs of size n, insertion sort runs in $8n^2$ steps, while merge ...
0
votes
1answer
27 views

Basic Logarithm equation, and how best to approach this question logically

Question: Solve the equation $$\log_3 \left(1 - 3x\right) = \log_9 \left(6x^{2} - 19x + 2 \right)$$ There's quite a bit going on, I'm trying to think about the best point to start in order to ...
0
votes
3answers
59 views

Changing the base of a logarithm

I must simplify $\log_4 (9) + \log_2 (3)$. I have tried but I can't get the correct answer $2 \log_2 (3)$. How do I proceed?
1
vote
1answer
17 views

on the convergence of an infinite series involving logarithms

It looks like the following quantity $$ q(k)=\frac{k+1}{2k}(1+\log k) - \sum_{i=2}^k \frac{i}{k^2} \log i $$ tends to $3/4$ as $k$ goes to infinity. Is there a nice way to prove it?
1
vote
1answer
41 views

Compute $(\ln(n!))^2$

In a discrete mathematics past paper, I must solve the following problem: We know (from the Stirling approximation) that ...
0
votes
1answer
31 views

Simplifying Logs

Simplify: $$\frac{\log a + \log b - \log c}{\log d^2}$$ Using the basic properties of logs, the numerator should simplify to $\log (ab/c)$, if I'm not mistaken. The denominator $\log d^2 = 2 \log d$ ...
4
votes
3answers
104 views

$ \frac{1}{2} + \dots + \frac{1}{n} \le \log n $

could anyone give me any hint how to prove this ? $$ \frac{1}{2} + \dots + \frac{1}{n} \le \log n $$ just came acroos this expression in my book.
1
vote
1answer
21 views

Finding a base given an exponent

In math, the logarithm of a number $n$ in base 10, finds the exponent where 10 has to be raised to, to produce $n$ again. So if $Log_{10}(n) = p$ then $10^p = n$. What I'm looking for is essentially ...
2
votes
2answers
44 views

How to prove this logarithm equation?

Given : $$\log_{12}18 = a \text{ and }\log_{24}54=b$$ prove that: $$ab + 5(a-b) = 1$$ My attempt: I couldn't solve it in any way, as base were not common. I could solve it if base of second ...
3
votes
2answers
25 views

Basic simultaneous equation with logarithms

Question Solve giving your answers as exact fractions, the simultaneous equations : $$8^y = 4^{2x + 3} \tag{1}$$ $$\log_2 y = \log_2x + 4 \tag{2}$$ I think that the RHS of eq 1 can be split up, ...
3
votes
3answers
49 views

Logarithmic Differentiation equation, Help!

So, I have to differentiate this via $\log$. I am still learning, so please be patient, I will try to explain everything I did. Please tell me if it is correct. ...
35
votes
3answers
806 views

Integrals of the form ${\large\int}_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx$

I'm interested in integrals of the form $$I(a,b)=\int_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx,\color{#808080}{\text{ for ...
0
votes
1answer
16 views

Binary Search 2Log(n)+1 steps?

So this is probably a basic and slightly stupid question. So.....for a binary search to find a number it takes at most 2Log(n)+1 steps (or Log(2N) questions. Im not a math major or anything, but ...
1
vote
2answers
51 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 ...
1
vote
0answers
35 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
votes
1answer
31 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 ...
0
votes
0answers
26 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 ...
-2
votes
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
votes
0answers
57 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 ...
1
vote
1answer
38 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
votes
2answers
81 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 $\log x =\log_e x$.
0
votes
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.
0
votes
1answer
37 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) = ...
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
33 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
26 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): ...
1
vote
3answers
42 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.
0
votes
3answers
39 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.
3
votes
5answers
208 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 ...
3
votes
2answers
44 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
87 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 ...
2
votes
6answers
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
votes
1answer
46 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
62 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
115 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 ...
2
votes
2answers
70 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 ...
1
vote
0answers
42 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 ...
0
votes
1answer
30 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?
0
votes
2answers
37 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
votes
2answers
27 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 ...