3
votes
1answer
53 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
57 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
47 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
26 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
64 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
62 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
55 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
59 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
38 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
20 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
19 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
14 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
24 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
38 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
95 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
60 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
46 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
24 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
32 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
2k 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 ...
3
votes
0answers
82 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
121 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
99 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
51 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
58 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
27 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
147 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
28 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
167 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
107 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
37 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, ...
2
votes
1answer
147 views

Number of digits and last digit of a number

How can I find the number of digits and the last digit of the number $$\large{2357^{2357^{.^{.^{.^{2357}}}}}}$$ Basically $2357$ to the power of $2357, 2357$ times.
2
votes
3answers
164 views

Can we *ever* use certain log/exp identities in the complex case?

This article on Wikipedia points out that certain identities for the log and exponential functions which are familiar from the real case require care when used in the complex case. Failures in the ...
2
votes
1answer
49 views

Derivative of $x\times n - 2^{\log_2 {x \times n}}$

I have a problem with solving derivative of $f(x)$ in this case: $$f(x) = x\times 10^9 - 2^{\log_{10} x\times 10^9}$$ This is what I have: $$f^\prime(x) = \lim_{m\to0} {f(x+m) - f(x)\over m}$$ $$= ...
0
votes
3answers
131 views

$1500=P \times { (1 + 0.02) }^{ 24 }$, what is the value of $P$?

Hey guys could you please tell me what is the faster why to solve this equation. It's a compound interest equation and I'm stuck at the ${ (1 + 0.02) }^{ 24 }$ I really don't know how to proceed in ...
2
votes
3answers
93 views

$a^b = c$, is it possible to express $b$ without logarithms?

$ a^b = c $ is it possible to express b without logarithms?
2
votes
2answers
97 views

rounding up to nearest square

Say I have x and want to round it up to the nearest square. How might I do that in a constant time manner? ie. $2^2$ is 4 and $3^2$ is 9. So I want a formula whereby f(x) = 9 when x is 5, 6, 7 or 8. ...
0
votes
1answer
35 views

Exponent, logarithmic question

I'm reading an article related to bioinformatics and I found this formula: Probability of $x =(1-y/n)^t$ or approximately $e^{-yt/n}$. My question is how do we pass to the approximation given in the ...
0
votes
2answers
77 views

$ 10^{-9}[2\times10^6 + 3^{1000}] $

$$ 10^{-9}[2\times10^6 + 3^{1000}] $$ I'm stuck on solving this. I wasn't able to put this into my calculator since the number is too big for it to calculate. So far I've done this: $$ ...
1
vote
2answers
93 views

discrete exponential calculating

I am interesting in about discrete exponential calculating. I know that $a^b = c\mod k$ is calculated as below. for example $3^4 = 13 \mod 17$. $3^4 = 81$; $81 \mod 17 = 13$. I am interesting ...
2
votes
3answers
78 views

Powers and the logarithm

By example: $4^{\log_2(n)}$ evaluates to $n^2$ $2^{\log_2(n)}$ evaluates to $n$ What is the rule behind this?
0
votes
2answers
276 views

basic math question: transform a sum of exponents to a sum of logarithms

I am sure this is a really dumb question but I am having trouble understanding it since I do not have any math background. I have the logarithms of 2 values: ...
1
vote
1answer
78 views

A simple inequality with logarithms and exponential

I want to prove that for $k>0$: $ 2^k \geq \frac{-1}{\log_2(1-\frac{1}{2^k})}$ I've plotted both functions and it seems to be the case for k>0. In fact, it would also be nice to see that: $ ...
1
vote
1answer
66 views

Trying to convert a nasty logarithm into an exponential

I have the following equation that I must express in terms of $r$: $$\Delta V = \frac{\lambda}{2 \pi \epsilon_0} \ln(\frac{r}{R})$$ This is a pretty tough one. I am not sure how to get the r out of ...
8
votes
5answers
226 views

Solving an equation with a logarithm in the exponent

I try to solve the following equation: $$ (N+1)^{\log_N{125}} = 216 $$ I know the answer is 5 here but how could I rewrite the equations so I can solve it? I tried to take the log of both sides but ...
0
votes
1answer
39 views

Logarithm exponent in Chernoff bound

I am applying Chernoff bound for a Poisson process with mean $\lg n$. I am putting $\delta =4$. Hence, $Pr(X<(1+4)\mu)< (\frac{e^\delta}{(1+4)^{(1+4)}})^\mu$ $ = (\frac{e^\delta}{5^5})^{\lg ...
5
votes
4answers
219 views

Can we prove $a^{\log_bn} = n^{\log_ba}$?

Can we prove $$a^{\log_bn} = n^{\log_ba}?$$ I forget how to prove this theorem. I picked up one numbers for test, and they worked.