Math Is Fun Forum

Scrabble has no replacement, and an easy modification to make to solve the problem you've pointed out, is to say:

'i' iterates over all 7 letter words with unique letter combinations:

Which would make POT TOP and OPT in your example treat as the same word since the use the same combination of letters.

or alternatively:

'i' iterates over all combinations of 7 letters that form valid words. Allowing repeated letters obviously.

simply because logarithms on the reals are not defined for negative numbers:

since ln(-n) is not defined for reals, neither is log_(-n)x although as you have shown there are special cases where you can define it

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if you venture into the complex realm, then you can define logarithms for negative bases:

natural logarithm of a negative real is given by the multivalued function

with principal branch:

so you could then have for some negative n

which gives for n = 5 and x = 25

and then, although this certainly is NOT '2' the expected answer, it is still of use since

Infact, you 'can' get 2 as the result if you do not use the principal branch, and instead define the logarithm as:

which gives

you can see this since:

although note that only the principal branch version is guaranteed to retain the property:

do you mean?

?

basicly, you have the line y = mx, and you have to find the values of 'm' such that every single point on the line y = mx, would be mapped to a point that is also on the line itself.

Does the symbol √ not refer specificaly to the 'principal' square root, aka the positive root, wherein simply 'square root' gives no such specifics, aka (x[sup]2[/sup])[sup]0.5[/sup] = ±x = ±√(x[sup]2[/sup])

I never even questioned that that bit could be wrong as it gave me the correct answer for the first part of the question I just need to be more careful in doing these things to make sure i don't leave myself prone to causing them.

although seemingly, using that equation involving k', i don't get 6.4 for v if plug in 150 for theta... what is wrong there?
seemingly there must be something wrong with the maths in my previous post.

I understand how you found T, but I still don't understand what is wrong with my method, i have integrated to get an equation for the centripetal force in terms of theta, which because of integrating has the constant which i find by plugging in a known pair for the centripetal force and theta.

from circular motion equations:

then potential energy:

the change in energy from increasing it's orbit then is the combination of the change in potential, and the change in kinetic

i'm really stumped how to get the answer they want

0/0 is equally undefined, who told you that it wasn't?

you're assuming ∞+1 =/= ∞, to show that ∞+1 =/= ∞, that is not allowed, i can equally show that pi = 1, by assuming pi = 1 to show that pi = 1, it shows nothing.