Write in logarithmic form :

$${3}^{4}=81$$

$${3}^{4}=81$$

gemuntertjx
2022-09-11
Answered

Write in logarithmic form :

$${3}^{4}=81$$

$${3}^{4}=81$$

You can still ask an expert for help

Aldo Harrington

Answered 2022-09-12
Author has **12** answers

$${3}^{4}=81$$

if a is a positive real number, other than 1

$${a}^{m}=x\Rightarrow {\mathrm{log}}_{a}x=m\phantom{\rule{0ex}{0ex}}{3}^{4}=81\phantom{\rule{0ex}{0ex}}{\mathrm{log}}_{3}81=4$$

if a is a positive real number, other than 1

$${a}^{m}=x\Rightarrow {\mathrm{log}}_{a}x=m\phantom{\rule{0ex}{0ex}}{3}^{4}=81\phantom{\rule{0ex}{0ex}}{\mathrm{log}}_{3}81=4$$

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asked 2022-07-07

Diffie hellman and the discrete algorithm problem

Suppose Alice and Bob are exchanging keys using Diffie-Hellman Key-Exchange Algorithm.

a - Alice secret key

g - generator

p - prime

x - the public key passed from Alice to Bob.

Eve is listening to the communication and she is exposed to the three parameters g,p,x.

She's using a brute-force method to find $a$, Alice's secret key.

Thus, Eve is looking for a satisfying this equation:

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Now, I know (By testing) Eve can find ${a}^{\prime}\ne a$ satisfying the equation above, and by using ${a}^{\prime}$ she can also compute the common secret key used by Alice and Bob.

Why is it mathematically true?

Suppose Alice and Bob are exchanging keys using Diffie-Hellman Key-Exchange Algorithm.

a - Alice secret key

g - generator

p - prime

x - the public key passed from Alice to Bob.

Eve is listening to the communication and she is exposed to the three parameters g,p,x.

She's using a brute-force method to find $a$, Alice's secret key.

Thus, Eve is looking for a satisfying this equation:

${g}^{a}modp=x$

Now, I know (By testing) Eve can find ${a}^{\prime}\ne a$ satisfying the equation above, and by using ${a}^{\prime}$ she can also compute the common secret key used by Alice and Bob.

Why is it mathematically true?

asked 2022-07-17

Is there a property for log(n)/n?

I found a small exercise which I couldn't figure what to do, so I found a solution. Then I tried to understand it and everything went well until I got to this part:

$\frac{1}{8}=\frac{\mathrm{log}(n)}{n}$

Then it just skipped and say that the answer was $n=43$. I was wondering if there is some kind of property for $\mathrm{log}(n)/n$ I don't know about. Or otherwise, how was this solved?

EDIT: This is the exercise Suppose we are comparing implementations of insertion sort and merge sort on the same machine. For inputs of size n, insertion sort runs in 8n2 steps, while merge sort runs in 64n lg n steps. For which values of n does insertion sort beat merge sort?

And this was the solution given:

$8{n}^{2}=64n\mathrm{log}(n)$

${n}^{2}=8n\mathrm{log}(n)$

$n=8\mathrm{log}(n)$

$\frac{1}{8}=\frac{log(n)}{n}$

I found a small exercise which I couldn't figure what to do, so I found a solution. Then I tried to understand it and everything went well until I got to this part:

$\frac{1}{8}=\frac{\mathrm{log}(n)}{n}$

Then it just skipped and say that the answer was $n=43$. I was wondering if there is some kind of property for $\mathrm{log}(n)/n$ I don't know about. Or otherwise, how was this solved?

EDIT: This is the exercise Suppose we are comparing implementations of insertion sort and merge sort on the same machine. For inputs of size n, insertion sort runs in 8n2 steps, while merge sort runs in 64n lg n steps. For which values of n does insertion sort beat merge sort?

And this was the solution given:

$8{n}^{2}=64n\mathrm{log}(n)$

${n}^{2}=8n\mathrm{log}(n)$

$n=8\mathrm{log}(n)$

$\frac{1}{8}=\frac{log(n)}{n}$

asked 2021-12-08

What's the correct notation for $\mathrm{log}$ squared:

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