Questions related to real and complex logarithms.

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3
votes
2answers
1k views

How to solve if I have ln on both sides of equation?

I thought this would be a common problem but googling hasn't helped. If I have $\ln(ex)=\ln(y) $ what the next step to solve for $y$?
2
votes
1answer
101 views

$\sum_{p\le x} \frac{1}{pq}$

I was given that $\sum_{p\le x} \frac{1}{p}$ = $\log\log x$+O(1). I need to show that $\sum_{pq\le x} \frac{1}{pq} = (\log \log x)^2 + O(\log \log x)$. Here we go: Break the sum into two sums: ...
5
votes
1answer
153 views

Derivative of $f(x)^{g(x)}$ at points when $f(x)=0$

I am interested in understanding the general behavior of the derivative for $$f(x)^{g(x)}$$ at points where $f(x)=0$. For example, if $f^g=x^n$ we have $$\frac{d}{dx}f^g(0)=\begin{cases}0 & n\ge ...
2
votes
4answers
85 views

Express $4\ln(x)+2\ln(x^4y^3)+5\ln(z)$ as a single logarithm

The problem is to express $4\ln(x)+2\ln(x^4y^3)+5\ln(z)$ as a single logarithm. Our teacher has shown us examples for the same base and when it's both add and subtract. But I'm not sure how to do ...
0
votes
2answers
53 views

Logarithmic equation

I'm studying logarithms and I encountered this equation: $$[\log_9(k+1)]^2+\log_9(k+1)+(k+1)>3$$ I tried a lot but I still couldn't solve it! I know this may be easy for most of you but please ...
-3
votes
1answer
37 views

How to solve for $k$ when the area about the $x$ axis and under the graph of the $f(x) = \frac1x$ from interval $x = [2, k]$ is equal to $\ln(4)$?

What approach would be ideal in solving for a number $k$ when the area about the $x$ axis and under the graph of the function $f(x) = \frac1x$ from interval $x = [2, k]$ is equal to $\ln(4)$?
2
votes
4answers
399 views

Prove that $\ln x \leq x - 1$

I need help with this proof for my real analysis class. it is on the practice sheets and we do NOT get an answer. I proved $\ln(x) < x−1$ for all $x>1$ by contradiction but cannot do this one. ...
6
votes
3answers
125 views

What is the value of $\ln \left(e^{2 \pi i}\right)$

I know that $$e^{2 \pi i} = 1$$ so by taking the natural logarithm on both sides $$\ln \left(e^{2 \pi i}\right)=\ln (1)=0$$ however, why isn't this $2 \pi i$ as expected? Is it beacuse logarithms ...
2
votes
2answers
39 views

Parametric inequation…

Supppose we have $a$ a real positive number that's not equal to $1$. Solve the following inequation: $$\log_a(x^2-3x)>\log_a(4x-x^2)$$ If it's known that $x=3.75$ is one solution of it.
2
votes
0answers
40 views

Yacov Perelman Nepero game

This is my first question, so sorry if I'll make any mistake in using the site formatting. I found this game on a book by Yacov Perelman and I thought it could be nice to introduce Nepero number to ...
2
votes
4answers
51 views

Why is $\log(n) \in o(\frac{n}{\log(n)})$?

This would be equal to: $\forall c>0: \exists n_0 \in \mathbb{N}: \forall n>n_0: c\log(n) ≤ \frac{n}{\log(n)}$ For $c=1$ this is obvious, because $\log(n) ≤ \sqrt{n} = \frac{n}{\sqrt{n}} ≤ ...
1
vote
1answer
82 views

Homework help to rearrange formula

Given the equation $${V_m} = u(\ln {m_0} - \ln {m_8}) - g{t_f}$$ I need to solve for ${m_0}$ Here is what I have but it looks messy and I feel like there is sometihng wrong or a better way 1st ...
4
votes
3answers
80 views

Why does $\int^{ab}_{a} \frac{1}{x} dx = \int^{b}_{1} \frac{1}{t} dt$?

I can't understand how the integral having limits from $a$ to $ab$ in Step 1 is equivalent to the integral having limits from $1$ to $b$. I'm a beginner here. Please explain in detail. ...
1
vote
1answer
65 views

Find $b-d$ when $\log_ab={3\over2}$ and $\log_cd={5\over4}$

$a,b,c$ are three natural numbers such that $\log_ab={3\over2}$ and $\log_cd={5\over4}$. Given: $a-c=9$ Find $b-d$
0
votes
2answers
28 views

Branch of logarithm which is real when z>0

I am familiar with the complex logarithm and its branches, but still this confuses me. I read this in a textbook: "For complex $z\neq 0, log(z)$ denotes that branch of the logarithm which is real ...
1
vote
1answer
27 views

Simplyfing Log Question

Don't know the concepts of diving and multiplying whole logs. $\frac{\log _a\left(x\right)}{\log _a\left(y\right)}\cdot \frac{\log _b\left(y\right)}{\log _b\left(x\right)}$ Can you please tell me ...
2
votes
2answers
42 views

Logarithm Evaluating

i'm new to this site and I need help on this logarithm question. I don't know how to approach this question to simplify it. $\frac{\log _2\left(81\right)}{2-\log _2\left(18\right)}$ Apparently the ...
1
vote
1answer
34 views

How does $10^{100}$ = $2^{\frac{100}{\log2}}$?

Googol is equal to $10^{100}$. To determine the number of bits that it needs to represented in binary, we need to rewrite Googol with a base of $2$. This is the correct answer: $$10^{100} = ...
1
vote
0answers
40 views

Find all the values of real parameter “n”…

Let $S$ be the set of real solutions for the following equation:$$\log_2(1-x-x^2)=n\log_{1-x-x^2}2+2$$ Determine all the values of real parameter $n$ for which $S\cap(0;{1\over2})\neq\emptyset$.
1
vote
6answers
19k views

Real life applications for logarithms [duplicate]

Can someone please tell me what purposes logarithms have in the everyday world? What non-theoretical applications are they in and when would one use them?
0
votes
1answer
37 views

Can't understand trivial discrete logarithm problem

I have a seemingly trivial problem with description: Find all discrete logarithms of base 2 of all non-zero elements in $Z_{11}$ field. I'm basing my learning on the notes I managed to grab ...
2
votes
1answer
30 views

Logarithmic problem with 2 variables help [closed]

How on earth do I solve? Any help will be much appreciated. The value of $M$ is given by $M = a \log_{10}S + b$. Note: Seismic moment measure the energy of the earthquake. Using the ...
0
votes
3answers
78 views

Limit of a sum of natural logarithms

As the title says I have to calculate a limit: $$\lim_{x\to0}\left(\ln(1+x)+\ln(1+2x)+...+\ln(1+px)\right)^x$$ I've transformed the sum into one logarithm $\ln\left((1+x)(1+2x)...\right)$etc but I ...
3
votes
1answer
80 views

On the equation $\exp(a x+b)=\ln(x)$

I am confronted with: $$\exp(a x+b)=\ln(x)$$ for $a,b$ reals and $a<0$, $b>0$. I need the (unique) solution for $x$. My first target is (if it exists) an analytic solution in terms of ...
0
votes
1answer
64 views

Solving the equation for $x$ given that $x \in \mathbb{R}, x > 0$

Given that the product of $\log(x+3)$ and $\log(x-3)$ is equal to $3$; the logarithms are to the base of 3. $$ \log_3(x+3)\log_3(x-3) = 3 $$ Someone to help me solve for $x$?
0
votes
0answers
59 views

Using spirals to draw log scale?

Taking inspiration from the fact every math textbook in existence puts a picture of a nautilus shell on its logarithms chapter, I did some research, and found that many different spirals (most easily ...
2
votes
0answers
74 views

Moving the branch cut of the complex logarithm

The complex logarithm is defined as $\log z:=\operatorname{Log} |z|+i\arg z$ , with the branch cut on the non-negative real axis. Determine a branch of $f(z)=\log(z^3-2)$ that is analytic at $z=0$ ...
1
vote
1answer
45 views

Solve $\log_2 (1+\frac{1}{x-1})<1$

I don't get how my teacher got two different equations out of the one. One is $> 0$ and the other one is $<2$. Be detailed please.
0
votes
1answer
62 views

Extraneous solutions where they come from?

I was doing some homework on logarithmic equations, and when I check my solutions on wolfram alpha I get that some aren't. So I'm interested in where do those extraneous equations come from?
0
votes
2answers
213 views

Different Polynomial Expansions of Natural Logarithm

I was recently Taylor-expanding ln around $(1,0)$. I noticed that this polynomial will have a range of input that converges between $0$ and $2$ regardless of Taylor ...
1
vote
1answer
43 views

Logistic Scoring Correction

"Consider the logistic curve $f(x)=\frac{1}{1+e^{-bx}}, -1 \leq x \leq 1$. We wish to use this curve to make a scoring correction formula $g(x)$ for an $n$ item test. The domain and range are both ...
1
vote
1answer
77 views

Conditions required for $(z_{1}z_{2})^{\omega}=z_{1}^{\omega}z_{2}^{\omega}$, where $z_{1},z_{2},\omega\in\mathbb{C}$

I am having trouble finding the conditions on $z_{1}$ and $z_{2}$ in order for: $$(z_{1}z_{2})^{\omega}\equiv z_{1}^{\omega}z_{2}^{\omega}\qquad \forall\omega\in\mathbb{C}$$ My first step was to ...
6
votes
3answers
114 views

Is this summation solvable? $S_n = \sum_{i = 2}^{n}\log_i{(n)}$

Is it possible to solve a summation with a variable base of log? $$ S_n = \sum_{i = 2}^{n}\log_i{(n)} $$ Should I use the derivative of $\log_i{(n)}$?
0
votes
1answer
45 views

Evaluate $\lim\limits_{x\to\infty}\frac{1}{\sqrt{x}}\int_1^x\ln(1+\frac{1}{\sqrt{t}})dt$

$\lim\limits_{x\to\infty}\frac{1}{\sqrt{x}}\displaystyle\int_1^x\ln(1+\frac{1}{\sqrt{t}})dt=?$ If the limit exists with l'Hopital i get ...
4
votes
2answers
112 views

Integral of $\frac{1}{x^2+1}$ using complex partial fractions.

Is there any way to evaluate the following integral via a complex partial fraction decomposition? $$ \int \dfrac{1}{x^2 + 1} \text{ d}x $$ So far I have: $$ \begin{aligned} \int \dfrac{1}{x^2 + 1} ...
1
vote
1answer
34 views

Insert Means in an Arithmetic Sequence (that contains logarithms)

So the question is: You have an Arithmetic Sequence. Log 2 and Log 1024 are two terms in the sequence Find 8 arithmetic means between them.
0
votes
2answers
44 views

What does $ \log_a (b) $ equal to?

Does $$ \log_a(b) = \frac{\log_c (b)}{\log_c (a)}$$ or $$ \log_a(b) = \frac{\ln (b)}{\ln (a)}$$ ?? Is there any difference between the two?
1
vote
3answers
63 views

Evaluate the log expression

Evaulate : $$ \frac{1}{\log_{xy} (xyz)} + \frac{1}{\log_{yz} (xyz)} + \frac{1}{\log_{zx} (xyz)} $$ I think that the following property of log will be used: $$ \log_a (b) * \log_b (c) * log_c (a) ...
0
votes
1answer
139 views

Dynamic Sizing of Circles Along a Logarithmic Spiral

I have created an logarithmic spiral in HTML canvas, and plotted circles along it. Using your mouse scroll wheel you can zoom in and out of the spiral (which works) – but I am having problems updating ...
0
votes
3answers
99 views

Find the largest possible root of a number that is whole

Using 8 as an example radicand, the degree would be 3 because ∜8 is not a whole number, while √8 is not the largest possible whole root. This type of problem is easy to calculate mentally with small ...
1
vote
3answers
78 views

Solution for $x$ with exponents?

I am trying to solve the following, $$7^{(2x+1)} + (2(3)^x) - 56 = 0$$ Should I put the 56 on the other side and get the log of both sides and is there a better way to solve this.
1
vote
1answer
53 views

reverse a logarithm

I have some data which produces the following logarithmic curve. As you can see, the curve produces the exact opposite of what Im trying to achieve (my data is the line with dots, the logarithm is the ...
0
votes
4answers
268 views

Solving a logarithmic system of equations

I am working on a test study guide and I can't seem to get the correct answer for this system of equations: \begin{align*} \ln(x) &= 3\ln(y) \\ \ 3^x &= 27^y \end{align*} I'm not ...
0
votes
2answers
55 views

Integral of $e^x ln(e^{2x} - 4)$

Find the integral from ln4 to ln6 of $$e^x \ln(e^{2x} - 4)$$ I factored $$\ln(e^{2x} - 4)$$ to get $$\ln((e^{x} - 2)(e^{x} + 2))$$ Then I separated this to get: $$e^x\ln(e^{x} - 2) + e^x\ln(e^{x} + ...
0
votes
2answers
46 views

Approximating the sum of integers with the logarithm

Why does the following hold? $\sum_{j=1}^{n-1}j \to \log(n) \text{ as } n \to \infty$ Thanks!
2
votes
0answers
18 views

Lipschitz continuity in two variables [duplicate]

Prove that $y \mapsto f(x,y)$ is Lipschitz continuous, where $$f(x,y) = \frac{y}{x} \ln{\frac{y}{x}}, \ \ \ |x-1| \leq \frac{1}{2}, |y-1| \leq \frac{1}{2}e$$ I tried to solve this, but I find it ...
0
votes
2answers
88 views

Is there a simple algorithm for exponentiating large numbers to large powers?

I've been thinking about this for some days, a multiplication is a lot of sums, so: $$75\times 75=\overbrace{75+75+75+75+75+75+75+75+\cdots}^{\text{75 times}}$$ But then, there is a simple algorithm ...
1
vote
3answers
53 views

Proof that $b^{\log_b(x)} = x$

I understand that the exponential functions are inverses, and would therefore map $x$ when formed as a composition, but I cannot find any formal mathmatical proofs. My thought process is: ...
0
votes
2answers
29 views

Find the inverse of the function

Find the inverse of the function $f(x) = -2 \cdot4^{2(x-3)} - 1$.
1
vote
1answer
53 views

probability: Chebyshev inequality question

For this question, I don't understand the highlighted part of the solution I thought it should be >5, but then 6?