Questions related to real and complex logarithms.

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0
votes
1answer
136 views

Undoing the Natural log after integrating $ln \frac{ \sec{x} \tan{x}}{3x+5}dx $

Since beating my head against a brick wall is so fun, I kept working on this old integral $\int \frac{ \sec{x} \tan{x}}{3x+5}dx$ . I think I have finally found a way to do it. Here goes. $$ \int ...
0
votes
2answers
25 views

Calculating pOH

If [OH-]=10^-pOH and [OH-]= 0.003 then what does pOH equal? I know this is simple but I just can't figure out how to do this calculation. Any help would be appreciated.
0
votes
2answers
68 views

Are the sums equal to each other?

They are $2$ different results for the integral $$\int xe^{2x}\sin\left(\frac x3\right)\,dx$$ $\displaystyle\frac{-3}{1369}e^{2x}\left(3(35-74x)\sin\left(\frac x3\right)+(37x-36)\cos\left(\frac ...
2
votes
2answers
310 views

Evaluating limit using logarithms.

Evaluate the following limit. $$ \lim_{x\to \infty} (\ln\ x)^{\frac{1}{x}} $$ What i have tried: $$ \ln\left[\lim_{x\to \infty} (\ln\ x)^{\frac{1}{x}}\right] $$ $$ \lim_{x\to \infty} \ln(\ln ...
1
vote
3answers
63 views

Estimating the natural logarithm from both sides: $1/(a+1)<\ln((1+a)/a) <1/a$

I must prove that for all $a>0$ $$\frac{1}{a+1}<\ln{\frac{1+a}{a}}<\frac{1}{a}$$ Can anyone help me?
1
vote
1answer
123 views

I wonder whether the system of equations and inequations below have a solution.

I wonder whether the system of equations and inequations below have a solution. If there are solutions, what are they? A numerical solution is also desired. $$\begin{cases} ...
0
votes
3answers
68 views

Logarithm properties doubt

The problem is $\log (5.64)^4$. According to the properties and laws of exponents, $\log (m^r) = r \log (m)$. But since the exponent is outside of the parenthesis in this problem, does it solves by ...
2
votes
2answers
419 views

How to take the Limits of Logs

How would you take the limit of $$\frac{\log(n!)}{\log(n^n)}$$ as $n\rightarrow\infty$. I believe you have to remove the log raising it to their base. Is this correct ? Thanks.
4
votes
1answer
198 views

Is $n^{\log c} = c^{\log n}$ true?

Is $n^{\log c}$ the same as $c^{\log n}$? If so, please explain.
0
votes
2answers
58 views

Absolute values in logarithms in a solution of differential equation

How have the moduli signs disappeared in the following step: $$\frac1{k}\left(\ln|g+kv| - \ln|g+ku|\right) = -t$$ Therefore $$ \ln\left(\frac{g+kv}{g+ku}\right) = -kt$$ $g$, $k$ and $u$ are ...
1
vote
1answer
286 views

Proving functions to be Big Oh

How do I determine if there exists a function $f$, such that \begin{equation} f(n) = {\mathcal O}(\log n), \end{equation} but \begin{equation} 2^{f(n)} ≠ {\mathcal O}(n). \end{equation} Is ...
0
votes
2answers
34 views

How do I calculate this logarithmic expression?

What I'm not sure about is the power of two above the logarithm. I just wanted to verify I'm calculating correctly Do I do these steps...? 1 - Take absolute value of variable AL 2 - Take log base ...
0
votes
1answer
37 views

Is there a seperate something in front logarithm that is raising a base to a power?

I am trying to solve a problem with the following form $$e^{\displaystyle A\log(x)}$$ $e^{\log(x)}$ is simply $x$, but how do I go about separating the $A$?
5
votes
3answers
101 views

What's a good class of functions for bounding/comparing ratios of complicated logarithms?

I have this goofy series $\sum \limits_{n=2}^\infty \frac{ \log_2 \left[ n \log_2^2 n \right]}{n \log_2^2 n}$ that Wolfram Alpha tells me diverges by the comparison test (and indeed, in the larger ...
1
vote
1answer
577 views

How to solve $\log \sqrt[3]{x} = \sqrt{\log x} ?$

How to solve $$\log \sqrt[3]{x} = \sqrt{\log x} $$
0
votes
1answer
52 views

How do I simplify and calculate this inequality?

$\log(x^3) > |x-1|$ I can't figure out how to go about solving this inequality, besides this one step: $3\log(x) > |x-1|$
1
vote
2answers
77 views

Solve the Logarithmic Equations for x, please.

This one is an exponential equation that I can't figure out.. $7^{x-2} = 5^{3-x}$ These two are logarithmic equations that I'm also having trouble with.. $\ln \sqrt[3]{x-6} = -2$ ...
0
votes
1answer
60 views

Prove if $f(x) = \ln\left(1-\frac{1}{x^2}\right)$ then $f(2)+f(3)+f(4)=\ln(5/8)$

I have that: $$f(x) = \ln\left(1-\frac{1}{x^2}\right)$$ I need to prove that $f(2)+f(3)+f(4)=\ln\left(\frac58\right)$ Indeed, I proved that $f(2)+f(3)+f(4)=\ln(3/4)+\ln(8/9)+\ln(15/16)$ But ...
3
votes
3answers
61 views

How do we prove that $a^{\log{b}} = b^{\log{a}}$ for $a > 1$ and $b > 1$?

I have tried using the change of base formula, but can't quite complete the equality: $$ a^{\log{b}} \\ a^{\frac{\log_a{b}}{\log_a{a}}} $$ How do I get the base of the exponent to be b?
1
vote
1answer
63 views

Growth of |logx| versus of 1/x

Do you think there is a number k s.t. $\int_{(0,\infty)} \frac{|log(x)|^{k}}{x}d\mu$ will converge,where $\mu$ is the Lebesgue measure? If you don't know ,can you at least give me some reference for ...
2
votes
3answers
95 views

$4\log_2(n)=n$ How to Solve for $n$?

As the title suggests, my log skills are pretty lacking. Need to learn how to get from $4\log_2(n)=n$ to $n=16$ ($\log$ base $2$). I've searched Google and it seems I am missing some core concept ...
3
votes
4answers
130 views

How can i calculate this limit

the limit is $$\lim_{x \to 0} {1 \over \ln(x+1)} - {1 \over x}$$ The problem is i don't know if i can calculate it normally like with a change of variables or not . Keep in mind that i'm not ...
1
vote
1answer
47 views

If$ a+b-1=1+\frac{ln(2^a-1)}{ln4}+\frac{ln(2^b-1)}{ln4}$ then $a=b$?

If $$a+b-1=1+\frac{ln(2^a-1)}{ln4}+\frac{ln(2^b-1)}{ln4}$$ where $a,b>0$ are real numbers and ln is $log_e$, then is a=b?
0
votes
1answer
26 views

Using log of function to determine orders of growth

If I have functions $f(n)$ and $g(n)$ and I would like to determine $f(n) \in \Omega g(n)$ and/or $f(n) \in \Theta(g(n)$. Does proving $\log(f(n)) \in \Omega \log(g(n))$ imply $f(n) \in \Omega g(n)$? ...
2
votes
2answers
165 views

How do you solve $x^{\log x}=100x$

How do you solve $x^{\log x}=100x$? Can you please thoroughly explain the left side of the equation. Please explain very clearly because I have only been learning logarithms for about a week.
2
votes
1answer
76 views

Find the Solution of the Exponential Equation?

How do I solve $5^x = 4^x+1$? I understand how to solve for $x$ when there is one exponent, but I don't know how to solve when there is an exponent on both sides of the equation..
1
vote
5answers
197 views

Proving $\log x^2 = 2\log x$

How does $\log x^2 = 2\log x$? Can you do a proof please. I know that this is true but I don't know why.
0
votes
2answers
77 views

Find $x$ in $\log x^2 = (\log x)^2$

Find $x$ in $\log x^2 = (\log x)^2$. I couldn't find x.
1
vote
2answers
60 views

Find the solution to this equation

The equation is the following $2^x-x^2-100=0$ It was handed to me today by a friend student of mine. I hope you enjoy it .
2
votes
2answers
464 views

Can I add $\log$ to both sides of inequality such way?

Can I add $\log$ to both sides of the following inequality $$f(n) \leq cn^k$$ and get $$\log (f(n)) \leq kc\log n$$ I know that by rules the result inequality should be like this $$\log(f(n)) \leq ...
-1
votes
4answers
1k views

Prove that $\log_2 3$ is irrational [closed]

Prove that $\log_2 3$ is irrational Seemingly simple homework assignment. Was never the best with logarithms, how would I go about proving?
2
votes
1answer
53 views

Simplifying an expression using a logarithm

I have the following expression $$\frac{1}{1+\rho}(1+n)^{(1-\sigma)}*(1+\gamma_{A})^{1-\sigma}<1$$ and have to use logarithms to get the following $$(1-\sigma)(n+\gamma_{A})<\rho$$ Could ...
0
votes
2answers
138 views

logarithm and exponent computation performance

Using glibc on a x86 processor, which takes more CPU time? $a\ log\ b$ or $b^a$? For which values of $a$ is one faster than the other? Optional: Does the base used matter? See also: What algorithm ...
5
votes
4answers
484 views

Proof $e^x = \exp(x)$?

Define $$\ln (x) = \int^{x}_{1}\frac{1}{t}$$ Assume I have proven that $\ln x$ is one-to-one and therefore has an inverse $\exp (x)$. Define $e$ as: $\ln e = 1$ Now, if you have no other notion ...
1
vote
3answers
73 views

How can you prove this equality?

I am trying to figure out the following equality, but cannot seem to get anywhere. I tried integrating by parts, but that blew up when I set u = (log x)^n and tried to take log (0). I also tried ...
11
votes
5answers
2k views

Why is it trivial that $\left(1+\frac{2\ln3}{3}\right)^{-3/2}\leq\frac{2}{3}$?

Can someone tell me why $$\left(1+\dfrac{2\ln3}{3}\right)^{-3/2}\leq\dfrac{2}{3}$$ is trivial because for me its not and I will need to do the calculation to see it.
2
votes
4answers
203 views

Showing $\frac{x}{1+x}<\log(1+x)<x$ for all $x>0$ using the mean value theorem

I want to show that $$\frac{x}{1+x}<\log(1+x)<x$$ for all $x>0$ using the mean value theorem. I tried to prove the two inequalities separately. $$\frac{x}{1+x}<\log(1+x) \Leftrightarrow ...
0
votes
1answer
87 views

Asset Depreciation - Logarithms

How many years would it take for the value of a car purchased at 30 000 to fall to 15 000 if it depreciates by 15% in the first year and 12% every year after that? This is not actually graded ...
1
vote
3answers
255 views

Derivative of $(\ln x)^{\ln x}$

How can I differentiate the following function? $$f(x)=(\ln x)^{\ln x}.$$ Is it a composition of functions? And if so, which functions? Thank you.
4
votes
8answers
589 views

Find $\lim_{x \to 0} \left( \frac{\ln (\cos x)}{x\sqrt {1 + x} - x} \right)$ efficiently

I need to evaluate: $$\lim_{x \to 0} \left( \frac{\ln (\cos x)}{x\sqrt {1 + x} - x} \right)$$ Now, it looked to me like a classic L'Hôpital's rule case. Indeed, I used it (twice), but then things ...
1
vote
0answers
40 views

Algebra of Exponential and Log Functions

This question may have a simple answer or a very complex one, but I am interested in what the reasons are for logarithms and exponential functions having the properties they have. To my knowledge ...
1
vote
4answers
111 views

Integrating $\ln(x)\times\ln(1-x)$

Is there a way I can derive the value of the integral $ \int_0^1 \ln(x)\ln(1-x)dx$ using the fact that $\displaystyle\sum_{i=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$ ? (the actual value of the ...
5
votes
2answers
108 views

Evaluate $\lim\limits_{y\to 0}\log(1+y)/y$ without LHR or Taylor series

Problem 12 on p. 297 of Spivak's Calculus (first edition) is Find $\lim\limits_{y\to 0}\log (1+y)/y$. (You can use L'Hospital's rule, but that would be silly.) I'm not sure the other method ...
4
votes
1answer
114 views

Proof of $e^{\ln(x)\ln(2)}$, which natural logarithm do I bring down?

I'm currently stumped with the proof for the following problem: $$F(x) = 2^{\ln(x)}$$ $$\Rightarrow F(x) = y$$ $$y = 2^{\ln(x)}$$ $$\ln(y) = \ln(2^{\ln(x)})$$ $$\ln(y) = \ln(x)\cdot\ln(2)$$ $$y = ...
0
votes
1answer
43 views

Proof of generalization of a particular limit converging to $e^{\frac{1}{(p-1)^2}}$

I was reading a very old and long article on logarithms in a library it has pages turned yellow and had one pages titled - Tricky problems I managed to solve 5 out of the 6 but I couldn't do this 6th ...
1
vote
3answers
150 views

I have a hard time understanding why $\ln e=1$

I have a hard time understanding why $\ln e=1$ Can someone explain to me why the natural logarithm of e is exactly equal to the first nonzero but positive integer?
0
votes
4answers
99 views

What calculate $\ln i$

I would like to know how to calculate $\ln i$. I found a formula on the internet that says $$\ln z=\ln|z|+i\text{Arg}(z)$$Then $|i|=1$ and $\text{Arg}(i)$ is?
0
votes
2answers
46 views

How do I evaluate this log expression?

Evaluate the expression $\log_8{8^{17}}$ I ended up getting $8^x = 8^{17}$. I'm guessing I find x, but that's a huge number, and I feel like I'm doing this wrong.
0
votes
3answers
46 views

Graphing $-\log_2x$ , $\log_2(-x)$, and $-\log_2(-x)$ by hand.

Trying to graph $-\log_2x$ , $\log_2(-x)$, and $-\log_2(-x)$. I don't understand which one reflects over the x axis, the y axis?
1
vote
1answer
68 views

Find the value of K. Use of l Hospital's rule and expansion is not allowed.

Let $f(x) =\log_{cos3x} (\cos2ix)$ if $x \ne 0$ and $f(0) = k$ where ($i$= iota) is continuous at $x = 0$, then find the value of $K$. Use of l Hospital's rule and expansion is not allowed.