Questions related to real and complex logarithms.

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20
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1answer
654 views

Integral $\int_0^\infty\frac{\ln\left(\sqrt{x+1\vphantom{x^0}}-1\right)\,\ln\left(\sqrt{x^{-1}+1}+1\right)}{(x+1)^{3/2}}dx$

Another integral similar to my previous question: $$\int_0^\infty\frac{\ln\left(\sqrt{x+1\vphantom{x^0}}-1\right)\,\ln\left(\sqrt{x^{-1}+1}+1\right)}{(x+1)^{3/2}}dx$$ Could you suggets how to evaluate ...
0
votes
2answers
51 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 ?
2
votes
1answer
2k views

Interpolation of a logarithmic function

I have a logarithmic function $$m \ln(x) + b$$ And three points $$(x_0, y_0), (x_1, y_1), (x_2, y_2)$$ The task is to find $m$ and $b$. Do I understand right that the third point is redundant? ...
1
vote
1answer
33 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 ...
0
votes
2answers
980 views

Graphing: Given two points on a graph, find the logarithmic function that passes through both.

Is there such a method to do this? I would like to come up with a logarithmic function (a graph that looks like a square root graph) that passes through two given points. Haven't had any luck in ...
1
vote
2answers
67 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 ...
0
votes
0answers
20 views

How to express $\log^p(n)$ where $p<0$?

I know $\log^4n=\log(\log(\log(\log(n))))$ and $(\log(n))^4=\log(n)\cdot\log(n)\cdot\log(n)\cdot\log(n)$ How do you express something like $\log^pn$ where $p<0$?
5
votes
1answer
46 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
vote
2answers
95 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
35 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
23 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
66 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 ...
0
votes
1answer
47 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 ...
1
vote
1answer
19 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 ...
3
votes
2answers
64 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 ...
1
vote
0answers
26 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,\ ...
4
votes
2answers
70 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 ...
2
votes
1answer
20 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 ...
3
votes
1answer
44 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 ...
0
votes
3answers
23 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 ...
5
votes
6answers
865 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 ...
2
votes
2answers
2k views

Why are logarithms not defined for 0 and negatives?

I can raise $0$ to the power of one, and I would get $0$. Also $-1$ to the power of $3$ would give me $-1$. I think only some logarithms (e.g log to the base $10$) aren't defined for $0$ and ...
0
votes
1answer
33 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
1answer
77 views

Hard question on logarithm [closed]

$$\log_7(\log_3(\log_2 x)) = 0$$ Find $x$. The solutions to the system of equations $$\left\{\begin{array}{cc}\log_{225} x + \log_{64} y &= 4\\ \log_x 255 - \log_y 64 &= ...
1
vote
1answer
148 views

CAGR, log of negative numbers

I am trying to calculate the Compound Annual Growth Rate (CAGR) of a number of companies using the Geometric mean of the annual growth of their EPS. Some of the EPS values are negative(loss making) ...
0
votes
3answers
48 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: $$(7x^6)(3x^2)$$
0
votes
2answers
604 views

Find Log equation from data points

I have the following data points, (left hand column goes from 0-127, right hand column goes from 30-22000 hz. Is there any calculator I can use to find a "log" function of this data, so that it comes ...
1
vote
1answer
47 views

Is it possible to have Logarithm with base 1 or 0?

I am wondering is there any definition that allows logarithm to have base 0 or 1 in real or complex fields (considering Euclidean space)?? Out-coming question is if you can define a logarithm with ...
0
votes
1answer
18 views

Some trouble with algebra using logarithms and summations

I'm having some embarrassing trouble with algebraic manipulation. I have the function $$f(y) = y^Tx-\log\sum_{i=1}^ne^{x_i}$$ and for each $i = 1,2,\ldots,n$ $$y_i = {e^{x_1} \over ...
6
votes
0answers
191 views

Two (strictly related) proofs by induction of inequalities.

Predictably, I'm stuck with the inductive steps. Let $p_m$ be the largest prime factor of $a_n$ and set $\lim_{n\to \infty}\frac{\log a_n}{p_m}=1$. Suppose also this ratio converges to $1$ faster than ...
1
vote
2answers
51 views

Must square root of e be positive?

I have always thought that there is two solutions to the square root of a real number, one being positive and the other being negative. However, in Penrose's book, A Road to Reality, he seems to claim ...
0
votes
1answer
22 views

Solve the below equation For $X$ in terms of $Y$ and $Z$ [closed]

Solve the below equation For $X$ in terms of $Y$ and $Z$ $XY-log(X)=Z$ Also give any other method like simulation, graph plot
2
votes
4answers
36 views

How can I differentiate this equation?

I need to differentiate this: $$ y = b(e^{ax}-e^{-ax}) $$ I've got the solution from a book, but I don't found the process to differentiate it. The solution is: $$ y = ab(e^{ax}+e^{-ax}) $$ Here ...
0
votes
1answer
259 views

Why are there two series representations of the natural logarithm?

On the Wikipedia article of the natural logarithm one finds two different series representations for $\ln(x)$: $\ln(x)= (x - 1) - \frac{(x-1) ^ 2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4} \cdots$ ...
1
vote
2answers
320 views

Logarithm question for Algebra 2/Trig class

$$\frac{1}{2} \log(x+2)=2$$ I'm decently good at logarithms but this one seems to be tricky, when I did it myself I got a negative decimal as my answer but I'm not 100% confident in it, and I would ...
0
votes
0answers
9 views

PDF of the logarithm of a chi-squared random variable

Could someone give me a hint, what could be the expression of the PDF of the following random variable Y: Y = a*log(b+X), where a,b are constants and X is a noncentral chi-squared distributed random ...
0
votes
2answers
40 views

How to differentiate $\ln(a^x)$?

Can someone give me the process to differentiate this (with respect to $x$)? $$ \ln(a^x) $$
8
votes
5answers
320 views

Natural logarithms base $e$

Why is $e$ used as a base of natural logarithms everywhere? Is the origin from the fact that exponential is the only function with the unique property of its differential and integral same and that ...
0
votes
2answers
38 views

Solving for $x$ using $\ln$ or any possible way.

$$ 12.46x=1-(1+x)^{-20} $$ I tried solving for $x$ using $\ln$ and other methods but the only answer i got was 0.8. The correct answer is approximately to $0.05$.
1
vote
1answer
52 views

Who are the two men credited with inventing logarithms?

This is a bonus question on a pre-calculus quiz I've been tasked with grading. Napier is clearly one of the answers. Who should I accept for the second inventor? In particular, should Newton be ...
5
votes
1answer
71 views

Lower estimate for $(\frac{\ln(1+2x)}{\ln(1+x)}-1)(1+2x)^{1/2}$ where $x>0$

I want to prove that: $$\left(\frac{\ln(1+2x)}{\ln(1+x)}-1\right)(1+2x)^\frac{1}{2}\geq 1$$ where $x>0$. Any help appreciated. Thanks!
2
votes
1answer
42 views

How I can solve this equation with respect to the variable $t$?

How I can solve this equation with respect to the variable $t$? $$\left\lfloor{\frac{\ln(t+1)}{\ln 2}}\right\rfloor=\left\lfloor{\frac{\ln t}{\ln2}}\right\rfloor+1$$ where $\left\lfloor {y} ...
2
votes
2answers
28 views

Solve the recurrence relation by taking the logarithm of both sides and making the substitution $b_n = \lg a_n$

Solve this recurrence relation: $$a_n = \left(\frac{a_{n-2}}{a_{n-1}}\right)^{\frac{1}{2}}$$ by taking the logarithm of both sides and making the substitution $$b_n = \lg a_n$$ A couple years ago ...
2
votes
1answer
32 views

How to get the peak value of this logarithmic equation?

Is there a way to get the peak point of the following equation? $$ (a_1-a_2 x)\ln\left(1+\frac{b_1 x}{b_2 x+b_3}\right),$$ where $a_1,a_2,b_1,b_2,b_3$ are all positive constant values and $x$ is also ...
1
vote
1answer
14 views

Is this change of variable correct?

Take the following function: $$\frac{dn}{d[\log(x)]} = a\exp{\frac{-(\log(x) - b)^2}{c}}$$ I'm interested in obtaining the form for $dn/dx$, so I take: $$\frac{d[\log(x)]}{dx} = \frac{1}{x} ...
2
votes
1answer
28 views

Cauchy Principal Value for log integral

How do I evaluate the expression $\lim_{\xi\to0}(\int_0^\xi\! \ln(\frac{1}{r})\frac{F}{\xi} \, \mathrm{d}r) $ , where$\ F,\xi $ are real numbers and $\xi\geq0$. Integration gives the expression ...
0
votes
0answers
38 views

Summation of logarithms

I am trying to calculate the sum $\ln(a-x_1)+\ln(a-x_2)+....+\ln(a-x_n)$ and solve it somehow with respect to a ($x_1,x_2,....,x_n$ are measurements of a simulation) . The number of terms in the sum ...
0
votes
1answer
29 views

Logarithm help - change to exponential form, solve.

I need help with number 55 and 57 here. I need to change to exponential form and solve, but I can't seem to figure them out. Thanks! Solve for $P$ $$\dfrac23\log R+0.05=\log P$$
0
votes
1answer
39 views

Complicated Logarithm

If $x>0$, $y>0$, and $$x^2 + y^2=98xy$$ then $\log(x+y)$ can be expressed as $A\log(x)+B\log(y)+C$ where $A,B,C$ are real numbers and all logarithms are base $10$ logarithms. Compute $100ABC$. ...