Questions related to real and complex logarithms.

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0
votes
2answers
61 views

Why does this inequality hold: $4n+2\le4n\log{n}+2n\log{n}$

Why is the following true? (I came across this in an algorithm analysis book but this inequality is not related to algorithm analysis) $$ 4n+2\le4n\log{n}+2n\log{n} $$
4
votes
4answers
240 views

Solve these equations simultaneously

Solve these equations simultaneously: $$\eqalign{ & {8^y} = {4^{2x + 3}} \cr & {\log _2}y = {\log _2}x + 4 \cr} $$ I simplified them first: $\eqalign{ & {2^{3y}} = ...
5
votes
4answers
3k views

log base 1 of 1

What is $\log(1)$ to the base of $1$? My teacher says it is $1$. I beg to differ, I think it can be all real numbers! i.e., $1^x = 1$, where $x\in \mathbb{R}$. So I was wondering where I have gone ...
-1
votes
1answer
88 views

$\log$ transform of the fundamental theorem of arithmetic? [closed]

Taking the canonical form of the fundamental theorem of arithmetic in the form: $$n=\prod_{j=1}^\infty p^{m_j}_j \qquad ;m_j\in \Bbb N_0$$ Does anybody know about a $\log n$ transform of this? Note: ...
3
votes
1answer
60 views

Logarithmic and exponential equation.

Find $x$ in $$\large3^{\log_{2} {x}} +3^{\log_{x} {2}}=90$$
1
vote
4answers
106 views

Solve this equation $\log_2x=\log_{5-x}3$

Solve this equation $$\displaystyle \log_2x=\log_{5-x}3$$ the answer is $x=2,x=3$ http://www.wolframalpha.com/input/?i=log_2%28x%29%3Dlog%285-x%2C3%29 Can you give me some hint
3
votes
1answer
379 views

integrating logarithm or x raised to a power?

$\int\frac{15}{x}dx$ would be 15$\int\frac{1}{x}dx$ = $15\ln|x|+c$. This seems like a silly question but I'm feeling exceptionally dense today. Why would you apply the logarithm rule, why wouldn't ...
2
votes
1answer
49 views

determining sign of function containing logarithm.

I would like to know the sign of the following term in general. I tried approximately and it was negative. Is there any $m_0$ such that for all $n>m>m_0$, the following function is positive or ...
0
votes
2answers
164 views

Arithmetic with the natural log

We have: $$ \ln(p^3 + 4) - \ln(4) = 2$$ What I did is: $$ \ln (p^3 + 4) = \ln(4) + \ln(e^2)$$ $$p^3 + 4 = 4 + e^2$$ $$ p = e^{2/3}$$ Why is this incorrect?
4
votes
1answer
162 views

How do you solve this equation: $10 = 2^x + x$?

Is it possible to solve this equation? \begin{align} a &= b^x + x \\ a-x &= b^x \\ \log_b(a-x) &= x \end{align} If $a$ and $b$ are known, how do you find $x$?
0
votes
3answers
49 views

$\log_x(y)=m^4$ and $\log_y(x)=5/(m^3)$, what is the value of $m$?

$\log_x(y)=m^4$ and $\log_y(x)=5/(m^3)$, what is the value of $m$?
1
vote
1answer
76 views

Making a logarithmic equation that starts at $(0,0)$ and passes through $(x, y)$?

I'm writing a computer program and for fading sound, it's best to do it in a logarithmic equation. What I need it to find a graph of the "volume" that starts at (0, 0) [x is the time, y is the volume] ...
0
votes
0answers
64 views

Expressing $\int _0^1da\int _0^{1-a}\ln ((a-1+b)^2-4 y a b )+\int_0^1 da\int _0^{1-a}\frac{(a-1+b)^2}{(a-1+b)^2-4 y a b}db$ in terms of dilogarithms

I came across these integrals and I'm trying to rewrite them in terms of Dilogarithms: $\mathrm{Li}_2(z):=-\int_0^z \frac{\mathrm{d}s}{s}\log(1-s)$. Can anyone suggest how to contunue? If there is a ...
7
votes
6answers
305 views

Elegant way to solve $n\log_2(n) \le 10^6$

I'm studying Tomas Cormen Algorithms book and solve tasks listed after each chapter. I'm curious about task 1-1. that is right after Chapter #1. The question is: what is the best way to solve: ...
3
votes
4answers
88 views

if $\log_xa+\log_ya=4\log_{xy}a$ prove that $x=y$

Let $x,y$ be numbers in the interval $(0,1)$ with the property that there exists a positive number $a\ne1$ such that $$\log_xa+\log_ya=4\log_{xy}a$$ I have used the property $$\log _ba=\frac{\log ...
3
votes
1answer
1k views

Integral proof of logarithm of a product property

In one of my textbooks, the expansion of a logarithm product is proved using integrals. $$\ln xy = \ln x + \ln y\iff \int_1^\left(xy\right)dt/t$$ $$\ = \int_1^xdt/t + \int_x^\left(xy\right)dt/t$$ ...
22
votes
1answer
249 views

What is a closed form of $\int_0^1\ln(-\ln x)\ \text{li}\ x\ dx$

Let $\operatorname{li} x$ denote the logarithmic integral: $$\operatorname{li} x=\int_0^x\frac{dt}{\ln t}.$$ Is it possible to find a closed form of the following integral? $$\int_0^1\ln(-\ln x) ...
2
votes
3answers
134 views

A tricky logarithms problem?

$ \log_{4n} 40 \sqrt{3} \ = \ \log_{3n} 45$. Find $n^3$. Any hints? Thanks!
3
votes
2answers
79 views

Solving the equation $10^{-x} = 5^{2x}$ with logarithms

$$10^{-x} = 5^{2x}$$ I'm having trouble isolating $x$. I get both logs on one side and then I'm stuck because I have nothing to divide with on the other side, and I can't factor it. Thanks
2
votes
2answers
74 views

Logarithmic Equations

How does one go about solving: $(5x+2)^{\frac{4}{3}} = 16$ I'm confused as how to parse through the equation to solve it using logs.
2
votes
2answers
53 views

Other log solutions?

I am evaluating the expression: $\ln(1)$ And I know the trivial solution is $0$. Are there other solutions to this equation? I feel there should be, my logic is as follows: if: $\ln(1) = x ...
6
votes
1answer
86 views

Sum of Logarithm Arguments

This is a very simple question I suspect but I just cannot seem to nail it... I have values for $X,Y,Z $, where $X =\log (x)$, $Y = \log (y)$ and $Z = \log (z)$ and I need to calculate $x + y + z$, ...
5
votes
3answers
83 views

Equation with Logarithm in Exponent

How to I solve the following exercise with a logarithm? I've forgotten the "trick" for doing so: $x^{log_{10} x} =10^4$
1
vote
1answer
189 views

Asymptotic analysis - if f(n) = Ω(g(n)), how to prove ln(f(n)) = Ω(ln(g(n)))?

Is the following statement true, if so, how can I prove it? if f(n) = Ω(g(n)), is also true that ln(f(n)) = Ω(ln(g(n)))? Since ...
2
votes
1answer
112 views

Natural logarithm

Can someone please suggest how one proves: $(1+2x)\ln(1+\frac{1}{x}) -2 >0$ where $x>0$. I plotted the function in a program and the inequality should be correct.
1
vote
2answers
48 views

simplifying equation with logs

I have the following equation: I would like to solve this for Ze. I have found the same equation expressed in terms of Ze in another paper: I can't get my head around how this works. This is my ...
1
vote
2answers
168 views

How do I divide a set of data samples which follow a logarithmic distribution?

I'm working for the first time with Logarithmic distribution. I have a set of samples which follow logarithmic distribution. I extracted the maximum and the minimum values from the set and defined the ...
3
votes
3answers
75 views

logs Challenge between two students >>be smart

two student were given the equation $2^{4x+6} = 3^{6x-3}$ 1.steve rearranged to get $2^{4x+6} - 3^{6x-3} =0$ then wrote $\log (2^{4x+6} - 3^{6x-3}) = \log0$ are these legal steps ? if not explain ...
1
vote
0answers
92 views

simplification of a natural log of a trigonometric function

hope you are all well. I am having a bit of a mental block, I am wondering if it is possible to simplify the following expression: $$k\cos X \cdot 4\ln(\cos X)$$ where $k$ is a constant and $X$ ...
1
vote
4answers
503 views

Logarithm question involving different base [closed]

Calculate the values of $z$ for which $\log_3 z = 4\log_z3$.
2
votes
3answers
98 views

$a^b = c$, is it possible to express $b$ without logarithms?

$ a^b = c $ is it possible to express b without logarithms?
35
votes
1answer
974 views

Closed form for $\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm dx$

I encountered this integral in my calculations: $$\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm ...
0
votes
3answers
2k views

On the pH scale, each unit change in pH represents a tenfold increase in acidity or alkalinity.

Trying to solve similar type equation to this. On the pH scale, each unit change in pH represents a tenfold increase in acidity or alkalinity. According to the diagram, vinegar is how many times as ...
1
vote
2answers
140 views

Floor of log equation $S=\left(\lfloor\log_{10}(x)\rfloor+1\right)x - \frac{10^{\lfloor\log_{10}(x)\rfloor+1}-10}{9}$

I must find 'x' and I don't know how to solve the following equation. Does it have a solution? How can I solve it? $$ S=\left(\lfloor\log_{10}(x)\rfloor+1\right)x - ...
17
votes
3answers
339 views

Closed form for n-th anti-derivative of $\log x$

Is it possible to write a closed-form expression with free variables $x, n$ representing the n-th anti-derivative of $\log x$?
1
vote
5answers
357 views

How to write in $2^x=5$ in logarithmic form?

How do I write: $$2^x = 5$$ In a logarithmic form? I've looked for a solution for some time now, so I decided to try here.
4
votes
4answers
292 views

Prove $\ln{(\frac {x}{y})} = \ln{x} - \ln{y}$ using the definition $\int_1^x\frac{1}{t}dt = \ln{x}$.

Prove $\ln (\frac{x}{y}) = \ln{x} - \ln{y}$ using the definition $\int_1^x\frac{1}{t}dt = \ln{x}$. I am able to prove $\ln{xy} = \ln{x} + \ln{y}$, and $\ln{x^r} = r\ln{x}$, but with this one, I am ...
2
votes
2answers
318 views

rounding up to nearest square

Say I have x and want to round it up to the nearest square. How might I do that in a constant time manner? ie. $2^2$ is 4 and $3^2$ is 9. So I want a formula whereby f(x) = 9 when x is 5, 6, 7 or 8. ...
4
votes
7answers
345 views

Solving $x^{\log(x)}=\frac{x^3}{100}$

How do I find the solution to: $$x^{\log(x)}=\frac{x^3}{100}$$ So I multiplied 100 both sides getting: $$100x^{\log(x)}=x^3$$ Now what should I do?
0
votes
2answers
98 views

Logarithmic equation. Need to know if i am teaching right

Two of my friends is studying for a test. They asked me about a simple question. But they told me that i was wrong on a question. I could be wrong. But i need you guys to make sure that they learn the ...
6
votes
2answers
342 views

Formula for Sum of Logarithms $\ln(n)^m$

As you know $\sum_{n=1}^k \ln(n) =\ln(k!)$ is there a formula for $\sum_{n=1}^k \ln(n)^m$?
0
votes
2answers
69 views

Solve a simple equation with log in it

I'm stuck with solving this equation, $$2 \log x = \log 9 $$ This is how far I made it: \begin{align} \log x &= \log 4,5 \\ x &= ? \end{align} I'm a beginner at logarithms so I appreciate ...
3
votes
4answers
976 views

Integrate by parts: $\int \ln (2x + 1) \, dx$

$$\eqalign{ & \int \ln (2x + 1) \, dx \cr & u = \ln (2x + 1) \cr & v = x \cr & {du \over dx} = {2 \over 2x + 1} \cr & {dv \over dx} = 1 \cr & \int \ln (2x ...
0
votes
1answer
60 views

Differentiate $y = \sqrt {{{1 + 2x} \over {1 - 2x}}} $ logarithmically

$\eqalign{ & y = \sqrt {{{1 + 2x} \over {1 - 2x}}} \cr & \ln y = {1 \over 2}\ln (1 + 2x) - {1 \over 2}\ln (1 - 2x) \cr & {1 \over y}{{dy} \over {dx}} = {1 \over 2} \times {2 ...
1
vote
3answers
87 views

Evaluating a limit with variable in the exponent

For $$\lim_{x \to \infty} \left(1- \frac{2}{x}\right)^{\dfrac{x}{2}}$$ I have to use the L'Hospital"s rule, right? So I get: $$\lim_{x \to \infty}\frac{x}{2} \log\left(1- \frac{2}{x}\right)$$ And ...
4
votes
4answers
153 views

Differentiate $\log_{10}x$

My attempt: $\eqalign{ & \log_{10}x = {{\ln x} \over {\ln 10}} \cr & u = \ln x \cr & v = \ln 10 \cr & {{du} \over {dx}} = {1 \over x} \cr & {{dv} \over {dx}} ...
4
votes
3answers
230 views

Write the expressoin in terms of $\log x$ and $\log y \log(\frac{x^3}{10y})$

What is the answer for this? Write the expression in terms of $\log x$ and $\log y$ $$\log\left(\dfrac{x^3}{10y}\right)$$ This is what I got out of the equation so far. the alternate form assuming ...
1
vote
2answers
106 views

Pre Calculus Math Equation With Logarithms

Please Help me with this I think i figured out question 1... but I get no solution... please help me start number 2 or if you can show full solution that be sick thanks. $\log_{3x}(81)=2$ ...
42
votes
2answers
1k views

Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$

Please help me to find a closed form for the following integral: $$\int_0^1\log\left(\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\right)\,{\mathrm d}x.$$ I was told it could be calculated in a ...
5
votes
2answers
151 views

How to formally show that $f(z)$ is analytic at $z=0$?

Let $z$ be a complex number. Let $$f(z)=\dfrac{1}{\frac{1}{z}+\ln(\frac{1}{z})}.$$ How to formally show that $f(z)$ is analytic at $z=0$? I know that for small $z$ we have ...