0
votes
2answers
41 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 ...
5
votes
1answer
291 views

Why does the method to find out log and cube roots work?

To find cube roots of any number with a simple calculator, the following method was given to us by our teacher, which is accurate to atleast one-tenths. 1)Take the number $X$, whose cube root needs ...
17
votes
10answers
2k views

Is there any simple method to calculate $\sqrt x$ without using logarithm

Suppose that we don't know logarithm, then how we would able to calculate $\sqrt x$, where $x$ is a real number? More generally, is there any algorithm to calculate $\sqrt [ n ]{ x } $ without using ...
2
votes
2answers
97 views

Alternatives to percentage for measuring relative difference?

If the value of something changes from $\;a\;$ to $\;b\;$, their relative difference can be expressed as a percentage: $$ \newcommand{\upto}{\mathop\nearrow} (D0) \;\;\; a \upto b \;=\; (b-a)/a ...
0
votes
1answer
55 views

Does $i = -\frac{(2\;W({\pi\over2}))}{\pi}$

Let $x = -\frac{(2\;W({\pi\over2}))}{\pi}$, where $W$ denotes the Lambert W-function. As $${\log(i^2)\over i} = \pi$$ and $${\log(x^2)\over x}=\pi$$ Does $x = i$?
0
votes
1answer
107 views

Smallest Mersenne prime with 100 million digits?

As some of you are probably aware, the Great Internet Mersenne Prime Search (GIMPS) is managing the search for the largest Mersenne primes of the form $M_p=2^p-1$, where $p$ is itself prime (GIMPS ...
1
vote
1answer
59 views

Calculating how long it will take for the half-life of X amount to fall below Y amount

I am trying to determine how long it will take ($t$) for the half life of 500 amount of substance to fall below 100 (to be $\le$ 99 -- I am only concerned with integers) when it has a half life of 5. ...
6
votes
3answers
257 views

Apparently cannot be solved using logarithms

This equation clearly cannot be solved using logarithms. $$3 + x = 2 (1.01^x)$$ Now it can be solved using a graphing calculator or a computer and the answer is $x = -1.0202$ and $x=568.2993$. But ...
0
votes
2answers
182 views

Logarithmic Number System Addition

Can somebody explain in simple terms how addition works in a logarithmic number system. Say I have the numbers A and B. These are logarithms (base e, say) of the actual quantities they represent. They ...
0
votes
1answer
128 views

Number of digits in base 10=9+1

Let $\tau:\mathbb N\to\mathbb N$ be the function that counts the number of digits of an nonnegative integer, i.e. $\tau(x)$ is the number of digits of $x$ in base 10. For example $\tau(5)=1$, ...
1
vote
1answer
69 views

Use identities to simplify $lg(a^2+b^2)^2$

Use logarithmic identities to simply the following: $$lg(a^2+b^2)^2$$ I started with \begin{eqnarray} lg(a^2+b^2)^2&=&2 \cdot lg(a^2+b^2) \\ \end{eqnarray} I think it's not the final ...
0
votes
2answers
110 views

How ${\left(\frac 12 \right)}^{{\lg n}}$ = ${\frac 1n}$ for any natural number $n$?

Consider binary logarithm . How is the value of ${(\frac 12 )}^{{\lg n}}$ = ${\frac 1n}$? I was going through this video of skiplists and the professor at 53:22 seconds make this claim .
1
vote
4answers
182 views

What is the $(\lg n)$-th root of $n$?

I am looking for the answer of the $(\lg n)$-th root of $n$, that is, $\sqrt[\lg n]{n}$. What is the answer and what log property should I use here? Please assume base as $2$ and $n$ as a natural ...
1
vote
3answers
763 views

arithmetic progression involving logarithm

$\log_2 X$, $\log_2 (X+9)$ and $\log_2(X+45)$ are 3 consecutive terms of an arithmetic progression; find $\qquad$(i) the value of X; $\qquad$(ii) the first term and the common difference; and ...
1
vote
1answer
104 views

Number of digits in different number systems?

I know a similar question was asked before, but I wanted to know if this can be extended to any number system by a generic formula. For example, given a number X in number system A, how many digits ...
2
votes
2answers
147 views

difference of logs

$\log(a - b) - \log(a - c)$ Does this have a simpler form? Perhaps one where the $a$s have cancelled out? I know it can also be expressed as a log of the fraction: $\log\frac{(a-b)}{(a-c)}$, but the ...