0
votes
1answer
38 views

Choosing a branch of the square root

Assume $O$ is the compliment of the non-positive part of the real line to the complex plane. This is an open and connected set. Only one of the values of $\sqrt z$ in $O$ has positive real part. With ...
0
votes
1answer
47 views

How to simplify this equation? $1+\sqrt {2^{2a_{n}+b_{n}+1}-16^{a_{n}}-4^{b_{n}}}=\log _{3}\left( a_{n}+b_{n}\right) $

How to simplify this equation? $1+\sqrt {2^{2a_{n}+b_{n}+1}-16^{a_{n}}-4^{b_{n}}}=\log _{3}\left( a_{n}+b_{n}\right) $
4
votes
5answers
69 views

What is the limit of $\log_k(k^a + k^b)$ for $k \to +\infty$?

I'm not very good with analysis (I never studied it) but because of my "work" on other topics of mathematics I came to this problem. $$\lim_{k \to +\infty }\log_k(k^a + k^b)=\max(a,b)$$ I'm sure ...
1
vote
2answers
49 views

Definition of $a^b$ for complex numbers

Problem statement Let $\Omega \subset C^*$ open and let $f:\Omega \to \mathbb C$ be a branch of logarithm, $b \in \mathbb C$, $a \in \Omega$. We define $a^b=e^{bf(a)}.$ $(i)$ Verify that if $b \in ...
1
vote
3answers
41 views

If $2 \cdot log_e{(x -2y)} = log_e{y} + log_e{x}$, then find the numerical value of $\frac{x}{y}$

If $2 \cdot log_e{(x -2y)} = log_e{y} + log_e{x}$, then find the numerical value of $\frac{x}{y}$ My try: $2 \cdot log_e{(x -2y)} = log_e{y} + log_e{x}$ $log_e{(x-2y)^2} = log_e{xy}$ ...
4
votes
3answers
358 views

Solving exponential equations using logarithms

This is the equation that I am having troubles with: $$\large x^{\large\log_{10}5}+5^{\large\log_{10}x}=50$$ So the first thing I do, I logarithm the whole expression with $\log_{10}$. So I get: ...
1
vote
1answer
40 views

Logarithmic Contest Question

The Problem was as follows: Define $\log*(n)$ to be the smallest number of times the log function must be iteratively applied to $n$ to get a result less than or equal to $1$. For example ...
0
votes
1answer
29 views

Exponential percentage decrease based on time

I have a bar that shows the time left for a task to finish and I want it to decrease faster as it gets closer to the end time. Example: Let's assume that the total time required for Task A to ...
1
vote
2answers
188 views

Properties of Logarithms

How do you simplify the following expression? $$\log\left(3^{(5^7)}\right)$$ I know that logarithms are like the inverse of exponents, but are there any tricks to simplify powers inside logarithms? ...
0
votes
0answers
12 views

Find the value of x if $\log_{2}({5 * 2^x + 1})$, $\log_{2}({2^{1 -x} + 1})$ and 1 are in arithmetic progression

My try: $1 - \log_{2}({2^{1 -x} + 1})$ = $ \log_{2}({2^{1 -x} + 1}) - \log_{2}({5 * 2^x + 1})$ $10 * 2^x + 2 = (2^{1 -x} + 1 )^2$ Let $2^x = t $ $10t^3 + t^2 -4t -4 = 0$ But I ...
1
vote
2answers
24 views

no. and nature of roots of $x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}} = \sqrt{2}$

The given equation is $$x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}} = \sqrt{2}$$ I took $\log_{2}{x}$ = $t$ and then rewrote the given equation as $$x^{3t^2 + 4t - 5} = \sqrt{2}$$ ...
0
votes
1answer
41 views

If $\log_{30}{3} = c$ and $\log_{30}{5} = d$ then the value of $\log_{30}{8} $ is??

I attempted the following: $\log_{30}{8} = 3\log_{30}{2}$ $\log_{30}{3} = c$ is equivalent to $3 = 30^c$ $\log_{30}{5} = d$ is equivalent to $5 = 30^d$ What should I do further?? Is ...
1
vote
5answers
109 views

solve the equation using logarithms (I think this is easy level)

Solve the equation for $x$ by using base 10 logarithms. $$16\cdot4^{2.5x}=9$$ EDIT: I made a typo (somehow... I was very far off!!) The correct equation is this: $$16\cdot4^{2.5x}=70$$ Can it be ...
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 ...
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 ...
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} = ...
3
votes
1answer
68 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 ...
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 ...
0
votes
2answers
53 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
1answer
37 views

weighted average with exponential weighting

I want to create weighted average, where weights depend on value of number. If I want exponential weights is this regular? $average = \log_e(\frac{\sum_{i=1}^n e^{v_i}}{n})$ Isn't it just average of ...
3
votes
2answers
71 views

Exponents with the same power

I've wanted to practice solving simple operations on exponents, so I've made a couple of equations to which I know the answers. $$5^x -4^x = 9$$ I feel really stupid, because I can't solve this one ...
0
votes
3answers
65 views

$(5^{2x}-1)(5^x)=1/5^x$ solve

I have the problem $(5^{2x}-1)(5^x) = 1/5^x$. I have already simplified it to $5^{3x}-1=1/5^x$ My question is when I do $\log$ base $5$ to the left side of the equation to get $3x-1$ by itself so ...
0
votes
0answers
63 views

Complex exponentiation

So I've got this question that is a bit difficult to ask, since it uses a term in my language that I can't properly translate into English. For $z\in\mathbb{C}^*$ and $a\in\mathbb{C}$ it would be ...
0
votes
1answer
101 views

Approximating Logs and Antilogs by hand

I have read through questions like Calculate logarithms by hand and and a section of the Feynman Lecture series which talks about calculation of logarithms. I have recognized neither of them useful ...
0
votes
1answer
42 views

show that $(1+ \frac {x}{n})^n < e^x$ and $e^x < (1- \frac{x}{n})^{-n}$ if $x<n$

If $n$ is a positive integer and if $x>0$,show that $(1+ \frac {x}{n})^n < e^x \quad$ and that $\quad e^x < (1- \frac{x}{n})^{-n} \quad $ if $x<n$ I proved the first one by the ...
-1
votes
1answer
27 views

Rewrite a formula in terms of exponential to the power of logarithm

I would like to rewrite the following formula, f(x). how can I rewrite the f(x) $$ f(x) = ...
1
vote
1answer
24 views

Comparing sum of fixed rate value to sum of escalating value

Find the number of years, $n$, until the sum of an escalating value/income exceeds the sum of a higher fixed level value/income. Income fixed at £8405.64 Income escalating @ 3% per annum from ...
0
votes
0answers
16 views

compounding interest question

A bank advertises that it compounds interest continuously and that it will double your money in 7 years. What is the annual interest rate? P(t) = P*e^kt P(t)/P =2 e^7k = 2 take ln of Both Sides ...
1
vote
2answers
25 views

Continuous compounding question

A population of rabbits starts out with $100$ rabbits. The growth rate is $11.7$% per day. Determine the exponential equation. Is it $$\mathbb {P(t)} = 100e^{11.7t}$$ Can you guys give me the ...
0
votes
1answer
43 views

Exponential of $\bar{z} $

I am currently reading the book Complex Variables by Stephen Fisher, there is one paragraph that was written like this: Establishing the following relation, and they write ...
2
votes
2answers
165 views

Solve exponential-polynomial equation

Solve the equation in $\mathbb{R}$ $$10^{-3}x^{\log_{10}x} + x(\log_{10}^2x - 2\log_{10} x) = x^2 + 3x$$ To be fair I wasn't able to make any progress. I tried using substitution for the ...
0
votes
3answers
62 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 ...
4
votes
1answer
57 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
1answer
27 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 $A$?
1
vote
0answers
35 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 ...
0
votes
1answer
41 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 ...
20
votes
5answers
3k views

Fun logarithm question

How would you go about solving $x^{2^{x}} = 2^{x}$? There should be a solution $1<x<2$, but I haven't found a way to derive the answer using the usual log laws, maybe there is an elegant way ...
4
votes
2answers
107 views

Limit of $x^x$ as $x$ tends to $0$

I am trying to solve the following limit: $$\lim \limits_{x\to0} x^x$$ The only thing that comes to mind is to write $x^x$ as $e^{x\ln{x}}$ and getting the right sided limit would be easy but I ...
8
votes
5answers
139 views

Solving equations of type $x^{1/n}=\log_{n} x$

First, I'm a new person on this site, so please correct me if I'm asking the question in a wrong way. I thought I'm not a big fan of maths, but recently I've stumbled upon one interesting fact, which ...
3
votes
4answers
101 views

Calculating $\log_7 125$

So the problem asks to calculate $\log_7 125$. It's multiple choice and the options are $2.48$ $4.75$ $1.77$ $2.09$ Given that $7^2 = 49$ and $7^3 = 343$, the answer must be either option 1 or 4, ...
0
votes
1answer
58 views

Solve some unusual log/exponential equations

I understand about log and exponential equations/functions, but I can't solve these (the numbers are just examples, of course): $ 4^x = x + 10$ $x^x = 3$ $(2x + 3x^2)^{x + 1} = (x - x^3)^{x^2}$ Are ...
3
votes
2answers
61 views

Is there a proof that $n^xm^x = (n^x)^{(\log(mn)/\log(n))}$?

This isn't a homework question, just something I'm curious about, but you can treat it that way if you like. So the other day I was playing with my calculator and I noticed that $$ 2^x10^x = ...
0
votes
1answer
29 views

Equations with exponents

I can't remember how to solve equations that have exponent and a variable in them. This is somewhat embarrassing, because this used to be really easy for me. I know that logarithms are involved I just ...
0
votes
1answer
29 views

Question about basic exponential/logarithm properties

Solve for $k$: $$e^{k/2}=a$$ Solution: $$e^{2k}=a$$ $$ k/2 = \mathbf{ln}a$$ $$ k=2\mathbf{ln}a$$ $$= \mathbf{ln}a^2$$ My question is: why does $2\mathbf{ln}a = \mathbf{ln}a^2$? Why can you ...
4
votes
3answers
162 views

$\log(n)$ is what power of $n$?

Sorry about asking such an elementary question, but I have been wondering about this exact definition for a while. What power of $n$ is $\log(n)$. I know that it is $n^\epsilon$ for a very small ...
0
votes
1answer
29 views

For each of series find the smallest $k$, that $a_n = O(n^k)$

Hey I need you to check my solutions: a) $a_n = (2n^{81.2}+3n^{45.1})/(4n^{23.3}+5n^{11.3})$ This one is done from $\sum_{i=1}^{k} O(a_i(n)) = O(max\lbrace a_i,..,a_k \rbrace )$ So it's ...
0
votes
3answers
279 views

Something to the power of a logarithm

This is probably a very obvious question, but here goes... An answer in my textbook claims that $$3^{\log n} = n^{\log 3}$$ and that $$4n^2 (3/4)^{\log n} = 4n^{\log 3}$$ Why, using more basic ...
0
votes
1answer
32 views

Growth rate of $n^2$ vs $(\log_3(n))^3$

Which grows faster, $n^2$ or $(\log_3(n))^3$? How do I figure out which grows faster in general in these kinds of situations?
0
votes
1answer
134 views

convert log(log(x)) to x-based power

I'd like to convert log(log(x)) to x-based power (I mean $x^{something}$). How can I do that?
2
votes
0answers
43 views

Can we define root extraction using Peano Arithmetic?

I've been playing with Peano Arithmetic and I've got multiplication, division, exponentiation, and logarithms. I can't figure out root extraction but I have a stab at it. Exponentiation: $a^0 = 1, ...