Questions related to real and complex logarithms.

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1answer
42 views

Solving logarithmic equation for $n$

I have the following equation and I am trying to isolate $n$: $$8n^2 = 64 n\log_2 n$$ Haven't done algebra in years and can't figure out how to get rid of the $\log_2$.
0
votes
1answer
25 views

How would you solve this logarithmic equation with both $\log n$ and $n$ terms?

I have a small logarithm related question that I do not seem to understand how to solve. If a program takes $\log n$ microseconds to run a program of size $n$, what is the maximum size of a program ...
1
vote
1answer
31 views

Can anyone help me in proving this?

As we know $e^{\ln (x)}=x$. To prove this I applied $\ln$ on LHS (left hand side) and I got RHS (rights hand side ) as follows $\ln (e^{\ln (x)})=\ln (x)$ $\ln (x)\ln (e)=\ln (x)$ $\ln (x)=\ln ...
1
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2answers
115 views

Intersection of functions $\ln(x)$ and $\frac{1}{x}$

How to find $x$ such that $$\ln(x)=\frac{1}{x}$$ Thank you!
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2answers
38 views

Investment question using logarithm

8,000 dollars is invested in an account that yields 6% interest per year. After how many years will the account be worth $14 000, to the nearest half year, if the interest is compounded monthly? ...
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2answers
71 views

Show that $(\ln a)^k \neq k \ln a $ [closed]

I have a question that I am not sure how to answer: Show that $(\ln a)^k \neq k \ln a $
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3answers
55 views

Solve logarithmic equation for $x$ to find the inverse of $f(x)= \ln(x+\sqrt{x^2+1})$

Let $f(x)= \ln(x+\sqrt{x^2+1})$. Find $f^{-1}(x)$. Here is what I got so far: $y= \ln(x+\sqrt{x^2+1})$, rewrite as $x= \ln(y+\sqrt{y^2+1})$, then $$e^x= y+\sqrt{y^2+1}$$ $$e^x-y= \sqrt{y^2+1}$$ ...
0
votes
1answer
37 views

How to compute antilogarithmic and superlogarithmic spaced values?

Let's suppose I have a range, e.g. $[100, 900]$. I want to compute 8 logarithmic spaced values $x_i={100, ..., 900}$. I use the following formula: $$x_1=\log(S)+\frac{(i-1)\log(S/L)}{n-1}$$ In the ...
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2answers
46 views

logarithm problems [closed]

How to carry out logarithm problems with a negative input?
6
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4answers
119 views

Are logarithms the only continuous function on $(0, \infty)$ that has this property?

Are logarithms the only continuous function on $(0, \infty)$ that has this property? $$ f(xy) = f(x) + f(y) $$ If so, how would we show that? If not, what else would we need to show that a function ...
1
vote
1answer
38 views

Analytical proof of the inequality $p^n (1-p\ln p)<1$ where $0<p<1$

Simulations with Mathematica suggest to me that $$p^n (1-p\ln p) <1\quad\text{for } p \in (0,1) \text{ and }n \ge 0$$ Have you got any hints on how I can derive an analytical proof of this ...
0
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0answers
32 views

On $p^{\log_q n}$, where $p$ and $q$ are distinct primes

Let $p,q$ be distinct primes, $n>1$ an integer with $\log_q n $ irrational. It was, and probably still is, a conjecture that $p^{\log_q n}$ is non-integer. What progress has been made towards it?
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4answers
51 views

help understanding how $\ln$ and $e$ cancel.

I realise cancel may be the wrong term and inverse may be more appropriate but these is one situation I really don't get…or rather haven't found a suitable explanation. Most sources I have come across ...
1
vote
0answers
21 views

Discrete logarithm - factorization of modul

I am solving discrete logarithm problem $mod N$. $N$ is composite number, i found its factors - lot of small primes and two big primes ($> 2^{50}$). Does the factorization of $N$ somehow help me? I ...
0
votes
1answer
24 views

Find roots for an equation with quadratic, linear and log terms?

I'm wondering if there exists a closed-form or analytic expression for the roots of an equation of the form $ax^2 + bx + c\log x=0.$ considering the natural $\log$. Wolfram alpha is leading me to ...
3
votes
2answers
91 views

Is it possible to solve this equation with logarithms and exponents?

$$-\frac{1}{3}\log(4x-12)+6=\left(-\frac{1}{2}\right)^x $$ Out of all the logarithm laws I've learned (which is pretty limited), I have not found a way to solve for what x is yet. Can someone verify ...
3
votes
1answer
60 views

Do these ratios of the Eulerian number triangle converge to the logarithm of x?

Consider the matrix $A_3$ with the definition if $n=k$ then $A_3(n,k)=\binom{n-1}{k-1}=1$, else if $n\ge k$ then $A_3(n,k)=\frac{\binom{n-1}{k-1}}{1-x}$ else $A_3(n,k)=0$. $\binom{n-1}{k-1}$ means the ...
0
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0answers
25 views

Properties of log map on matrices in $SE(3)$

I am learning about the log map on $SE(3)$ and I want to check my understanding of properties for use in solving an equation. Are the following true, for A, B, C as elements of $SE(3)$? $$ \log(ABC) ...
-1
votes
1answer
44 views

Solving a logarithmic inequality

For what values of $x$ it holds that: $$-\log_{e}(1-x) \geq x$$ and how can we prove this? $\log_{e}$ is the natural logarithm $\ln$.
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0answers
16 views

Fisher Transform: Use of the natural logarithm of negative number. How is it possible?

I have the following equation, from http://www.mesasoftware.com/Papers/USING%20THE%20FISHER%20TRANSFORM.pdf Because the parameters inside the log are (1+x)/(1-x), the output is always negative when ...
3
votes
2answers
92 views

Find $\frac{1}{\log 2}+\frac{1}{(\log 2)(\log 3)}+\frac{1}{(\log 2)(\log 3)(\log 4)}+ \cdots$

Is it possible to calculate the sum of $\dfrac{1}{\log 2}+\dfrac{1}{(\log 2)(\log 3)}+\dfrac{1}{(\log 2)(\log 3)(\log 4)}+ \cdots$?
0
votes
1answer
69 views

solving mod equation

I am attempting to solve $r_1$ in this equation: $$m + xr \equiv m_1 + xr_1 \pmod q$$ This is what I derived at: $$m-m_1 + xr / x \equiv r_1 \pmod q$$ I proceed to sub these with the necessary ...
0
votes
1answer
47 views

Maximize ratio of logarithms

How can one maximize the ratio of two logarithms $ \frac{\log{f(x)}}{\log{g(x)}}$ where the argument to each logarithm is the (positive) ratio of two first-degree polynomials? I have tried ...
3
votes
2answers
103 views

Proof of a closed form of $\int_0^1(-\ln x)^ndx$

$$\int_0^1(-\ln x)^ndx$$ Is there a step-by-step solution to a closed form of this expression? I've tried using different representations to re-write the expression but I couldn't find anything I knew ...
2
votes
3answers
126 views

Show without differentiation that $\frac {\ln{n}}{\sqrt{n+1}}$ is decreasing

Show that the function $\displaystyle \frac {\ln{n}}{\sqrt{n+1}}$ is decreasing from some $n_0$ My try: $\displaystyle a_{n+1}=\frac{\ln{(n+1)}}{\sqrt{n+2}}\le ...
2
votes
2answers
82 views

Show without derivative that function $\frac{\ln{n}}{ n\ln{\ln{n}}}$ is decreasing

I have a problem with showing the function $\displaystyle \frac{\ln{n}}{n \ln{\ln{n}}}$ is decreasing. I came to form $(n+1)^{\ln{\ln{n}}}<(n)^{\ln{\ln{(n+1)}}}$ and I don't know how to show that ...
2
votes
2answers
124 views

Solving the equation $\ln(x)=-x$

I tried solving this equation for a long time but did not succeed. Any help is appreciated. $$\ln x=-x$$ I am not sure the tag is correct, I am not familiar with English mathematical terms. Please ...
6
votes
1answer
107 views

$\log^2 (x^2) + \log (x-1) = 0$

I'm trying to solve the equation $\log^2 (x^2) + \log (x-1) = 0$ but all I could do is to show that $1 < x < 2$. Wolfram Alpha says that $x = 1.508554...$, this is good, but I really want to ...
2
votes
2answers
55 views

Upperbound this difference between two log expressions

I have the difference between the following log expressions and I am trying to bound the difference, $$F= \log \left(1+ \left(2+\frac{1}{\sqrt{2}}\right)^2 x^2\right) - \log \left(1+ ...
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votes
0answers
35 views

Transforming a logarithmic expression?

Do you know any nice way to rewrite $\log(1-e^{A})$?
3
votes
8answers
298 views

Solution of an exponential equation

Probably very simple question. Why the solution of $$1=n(1-a)^{t}$$ in terms of $t$ is equal to: $$t=\frac{\ln n}{\ln \frac{1}{1-a}}$$
1
vote
1answer
40 views

How do I prove this derivation?

I hope you can help me with this one because I seem to not quiet get a start here :/ Lets say we got a $b\in\mathbb{R}_{\gt 0}$ and a $y\in\mathbb{R}$ and we define $b^y:=\exp\left(\ln b \cdot ...
1
vote
2answers
69 views

Why does this inequality stand?

I want to ask something about: "Since $i \log_e i$ is concave upwards, it is easy to show that $$\sum_{i=2}^{n-1} i \log_e i \leq \int_2^n x \log_e x \,dx \leq \frac{n^2 \log_e ...
5
votes
1answer
48 views

How to prove that $f(x) - f(x-1)$ approaches $\frac{\log_{10}(10)}{\log_{10}(e)}$?

Let $$f(x) = \sum_{n=1}^{10^x}\frac{1}{n}$$ I noticed that as x approaches $\infty$, $f(x) - f(x - 1) \approx 2.3025$. After a bit of experimenting, I found that $2.3025... = ...
1
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2answers
100 views

How does $n < 2^n$ become $\log n < n$ by taking log of both sides?

How does $n < 2^n$ become $\log n < n$ by taking the log of both sides? I understand the left side but I do not understand the right side of the inequality. The once was $\log 2^n$ becomes $n$ ...
2
votes
1answer
41 views

Find the leftmost (most significant digits) of a large exponent calculation, say $99^{99}$

I want to find the initial 10 digits of an exponent calculation whose result is a very large number - Say, $99^{99} = 3.697296 \times 10^{197}$ I only need to know the digits $3697296$ Is there any ...
0
votes
1answer
26 views

Problem on logarithms

if lg2=x, lg3=y ,then i) 2/9 ii) 75 iii) 0.0015 Write logarithm base 10 of x and y Please help me to resolve this problem. For first one I got this, Is this correct? 10^x=2 10^y=3 =10^x / (10^y)^2 ...
7
votes
1answer
72 views

$a_{n+1}=\log(1+a_n),~a_1>0$. Then find $\lim_{n \rightarrow \infty} n \cdot a_n$

Suppose that $a_{n+1}=\log(1+a_n),~a_1>0$. Then find $\lim_{n \rightarrow \infty} n \cdot a_n$. I can find $\lim_{n \rightarrow \infty}a_n=0$. But I have no idea to find $\lim_{n \rightarrow ...
1
vote
1answer
23 views

Double solutions and plotting transcendental equations

I have the following transcendental equation: $y^2 - \log(y)^2 = 4\cdot\log(x) + 4/x + C$ and I aim to plot the equation in the positive, real quadrant, with $x>0$ (actually in the $0 < x ...
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vote
2answers
65 views

Reverse an equation with ln and power

I'm trying to solve for $x$ in the following equation: $\ln(y) = a \cdot (\ln(x)) ^ b + c$ $a = 0.0838 b = 2.6275 c = 0.2506$ but my results look bad. Can anybody show me his demonstration ?
0
votes
1answer
57 views

Reasoning behind multiplying by conjugates

What is the reason behind multiplying by conjugates? I am currently studying single variable calculus and throughout the lessons from the text I'm using, the reasoning as to why one would multiply by ...
3
votes
2answers
72 views

How many numbers less than $x$ have a prime factor that is not $2$ or $3$

I am trying to figure out the number of integers greater than $1$ and less than or equal to $x$ that have a prime factor other than $2$ or $3$. For example, there are only two such integer less than ...
2
votes
0answers
34 views

Existence and uniqueness of a function generalizing a finite sum of powers of logarithms

I hope to find a proof of the following conjecture: $(1)$ For every $a>0$ there is a convex analytic function $f_a:\mathbb R^+\to\mathbb R$ such that: $f(1)=0$ and $\forall x>1,\ ...
2
votes
1answer
21 views

Discrete logarithm when $\alpha$ is not a primitve root

When a number $\alpha$ is a primitive root for a prime number $n$ then $\beta \equiv \alpha^{x} \mod n$ can be written as $x = \log_\alpha(\beta) \mod n-1 $. If $n$ is not a prime, the equation ...
0
votes
3answers
25 views

Limit and L'Hopitals

I'm having trouble with this problem. $\lim{n \to \infty} (1+\frac 3n)^n$ My professor said to use a proof to figure out that the limit of the ln of the function is 3, but I can't figure out how to ...
3
votes
1answer
50 views

Proving analytic continuation, choosing suitable branch cuts,

Consider the function $$f(z)=\log[(z^2+1)^{1/2}],\quad z>0$$ where the branch is chosen so that $(z^2+1)^{1/2}>0$ for $z>0$ and the log denotes the principal branch. Let $R$ be the union of ...
4
votes
2answers
75 views

How to integrate $\ln \big( b + \sqrt{b^2 + c^2 + x^2}\,\big)$?

I am looking to demonstrate the following result. Any ideas are much appreciated. $$ \begin{align}\int \ln \left( b + \sqrt{b^2 + c^2 + x^2}\right) dx = &\;x \ln \left( b + \sqrt{b^2 +c^2 ...
5
votes
6answers
894 views

Alternate proof for “$\log_{10}{2}$ is irrational”

I need to prove that $\log_{10}{2}$ is irrational. I understand the way this proof was done using contradiction to show that the even LHS does not equal the odd RHS, but I did it a different way and ...
0
votes
1answer
40 views

Interpret a linear scale as a logarithmic scale

Answer The solution to my question is $10^{\log{logMin}+\frac{x-linMin}{linMax-linMin}(\log{logMax}−\log{logMin})}$ with $x$ being a value on a linear scale $linMin$ to $linMax$ that is being ...
0
votes
3answers
53 views

Find the derivative of $F(x) = x^7 \ln(x^3 e^{3x^2 -8})$

Find $F'(x)$ for $$F(x) = x^7 \ln(x^3 e^{3x^2 -8})$$ Here is what I have so far: $$(7x^6)(3x^2)$$