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solve for x (3 dp) a) 3^x=19. b) 2^1-x=7 c) (1/4)^x=75. How to do this?
You need to use logarithms
a) if 3 ^ x = 19 x = log 19 to the base 3 x = 2.68

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for b) 2^1 = 2 2 - x =7 -x = 7-2 x = -5
for c) if (1/4) ^ x = 75 x = log 75 to the base 0.25 x = -3.11
I think for b, I got 1.807.. and for a and c I got the same answer as yours.
your answer gets to 7.28 My answer gets to 18.99 To check your answer for part b), substitute it in 3^x = 19
the formula is if x^y = z y = log z to the base x
hm.. ok.
You understand logarithms right
hm.. yup I understand it.
So then you should show this as your method
adam an eve take turns flipping a coin. the first one to flip 'heads' wins. if eve goes first, what it the probability she wins?
1/2 ?
nope. she has the advantage of going first, so it must be greater than 1/2.
I dont think so The probability of her getting heads is half The probability of Adam getting heads and her cetting tails is 0.5 * 0.5 thats 1/4 So she should be 1 - 0.25 = 0.75
no that is not it either but it is getting closer. two methods (which really amount to the same thing) a linear equation, or write out sample space of 'eve wins' and sum the geometric series.
No i think it should just be 0.75 I can't think of anything else
Whats your answer and method
how could she win? get heads on first try, done. i represent as (h) and probability is 1/2 eve gets tails, adam gets tails, and then eve gets heads. represent as (t t h) and probability is 1/8 eve gets tails, adam gets tails, eve gets tails, adam gets tails, eve gets heads (t t t t h) probability is 1/32 etc these events are mutually exclusive, so add them up \[\frac{1}{2}+\frac{1}{8} +\frac{1}{32}+...\]
that is method one. method 2: put P = probability eve wins. write a linear equation in P as follows. she could win on first try with probability 1/2 OR she flips tails, adam flips tails [this happens with probability 1/4], AND she gets another turn. now if she gets another turn that is just like starting the game over, so her probability of winning has not changed. OR means we add the probabilities, AND means we multiply (assuming they are independent, which they are) and so we get an equation in P \[P = \frac{1}{2} + \frac{1}{4}P\] If you sum the geometric series you see that it is identical so solving the linear equation for P

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