Questions related to real and complex logarithms.

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1
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5answers
90 views

What is the derivation for the derivative of $a^{t}$

Been driving me nuts. Can someone prove to me that $$\frac{d(a^t)}{dt} = a^t \ln(a)$$ Thank you!
3
votes
3answers
133 views

Logarithm base transformation

I am trying to solve a problem which, I think, revolves around base transformation of logarithms. It goes like this: $\log_5\,{\log_6\,{\frac{6x-1}{x+1}}} < ...
1
vote
2answers
19 views

Logarithms involving decimals

I am a student wondering how would I put this correctly into a calculator. I have 1,05 and 1,216 1,05^n=1,216 How would I calculate n without just multiplying 1,05 against itself until I hit the ...
1
vote
2answers
29 views

Function application (word problem)

The problem: My work so far: $3=log(\frac{A}{A_0})$--->$10^3=\frac{A}{A_0}$ $\frac{A}{A_0}=1000$ (Am I done there?) Plugging it in: $M=log(\frac{1900000}{1000})$ $10^M = \frac{1900000}{1000}$ ...
2
votes
2answers
33 views

Extra Payment Mortgage Calculator

Mortgage formula I'm using: $$M = P \left(i + \frac{i}{(1+i)^n-1}\right)$$ where $M =$ payment amount, $P =$ principle balance, $i =$ term interest rate, and $n =$ number of terms. But now I'm ...
7
votes
4answers
361 views

Hints on calculating the integral $\int_0^1\frac{x^{19}-1}{\ln x}\,dx$

I would be happy to get some hints on the following integral: $$ \int_0^1\frac{x^{19}-1}{\ln x}\,dx $$
1
vote
1answer
26 views

I need help with Expanding logs

I know the log rules for expanding but I am not sure how to expand these difficult ones: (also the x's here depending upon the context are multiplication) $$\log_4(x^4yz)^2$$ $$\log_3 (((6\times ...
2
votes
2answers
57 views

Logarithm question-I donot know but this question may be solved by any other way also.

Let $(x_0,y_0)$ be the solution of the following equations. $$(2x)^{\ln{2}}=(3y)^{\ln{3}}$$ $$3^{\ln{x}}=2^{\ln{y}}$$ Then $x_0$ is A) $\frac{1}{6}$ B) $\frac{1}{3}$ C) $\frac{1}{2}$ D) $6$ I ...
0
votes
2answers
44 views

Does the logarithm inequality extend to the complex plane?

For estimates, the inequality $\log(y)\le y-1,$ $y>0$ is often helpful. Is there any sort of upper bound for the logarithm function in the complex plane? Specifically, $|\log(z)|\le$ something for ...
0
votes
1answer
59 views

Finding time constants of a circuit?

So this is a homework question and I am having trouble figuring out what they are asking. 'The potential difference (voltage) across the capacitor at time t > 0 is given by $V_C(t) = q(t)/C$. The ...
3
votes
2answers
54 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 $y$?
1
vote
1answer
86 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
144 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 ...
0
votes
2answers
48 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 ...
-2
votes
1answer
33 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 $\ln(4)$?
1
vote
4answers
87 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
96 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
26 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
26 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
43 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}} ≤ ...
0
votes
0answers
13 views

Logarithmic and Linear Growth

Imagine I have an equation: $L=f(x)-(a(x)-b(x)+c(x)-...q(x))$ Where $f(x)$ grows at a linear rate and $a(x)$ through $q(x)$ each independently grow at a logarithmic rate and $q(x)<log_2(x)$. ...
0
votes
1answer
42 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 ...
5
votes
3answers
64 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
64 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
21 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
17 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
27 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
24 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
24 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
322 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
26 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
25 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 ...
1
vote
3answers
48 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
71 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
18 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
38 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
39 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
43 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 ...
0
votes
1answer
35 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
64 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
2answers
71 views

Is this summation solvable? $S_n = \sum_{i = 1}^{n}\log_i{(n)}$

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
41 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
75 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
12 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
34 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?
0
votes
3answers
54 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
40 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
40 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 ...