Question

someone please help me
fan, medal, testimony

Answer 1

what do you multiply by to get from the number of runs in year 1 to # in year 2 ?

Answer 2

multiply by 2? @phi

Answer 3

yes. when you multiply (like in this problem) to get to the next number, that means think exponents

the multiply by 2 should be the hint:
2
2*2
2*2*2

are the first 3 entries.
match those with the year
1 2
2 2*2
3 2*2*2

do you see a pattern ?

Answer 4

not really ): that was kinda confusing. ):

Answer 5

for year 4, how many runs do you expect to see (based on the pattern) ?

Answer 6

1 2
2 2*2
3 2*2*2
4 ?

Answer 7

2*2*2*2

Answer 8

yes, and can you exponents to write the "short way" ?

Answer 9

if you multiply something by itself 4 times
2*2*2*2
then using exponents (because people wanted a short way to write this)
\[ 2^4 \]

Answer 10

so your table could be written like this
\[ 1 \ 2^1 \\ 2\ 2^2 \\ 3\ 2^3 \\ 4\ 2^4 \]
or if you use "x" as the year
\[ x \ 2^x \]

Answer 11

for year "x", the number of runs will be
\[ runs= 2^x \]

Answer 12

I understand now(: but how would I be able to put that in words? to complete the question. the only part im confused about now, is the number I am given (256)

Answer 13

to "explain how to create an exponential function" I would say
number of runs for year x is 2^x, so the exponential function is
r = 2^x
where x is the year and r is the number of runs

to convert to a "logarithmic function", take the logarithm of both sides
log_2(r) = x

or if we use log base e
ln(r) = x ln(2)
\[ \frac{\ln(r)}{\ln(x)} =x \]

Answer 14

to solve for the year when r = 256
we can take the log base 2 of 256
\[ x= \log_2(256) \]

if you know your powers of 2 you can answer that.
but if you need a calculator, you need to change to base 10 or base e
\[ x = \frac{\ln(256)}{\ln(2)} \]

Answer 15

wait, so all I do is plug 256/2 into the calculator? @phi

Answer 16

no. unfortunately logs are confusing
what you are trying to figure out is how many 2's to multiply to get 256
2*2*2*2*2...*2
how many ?
one way is keep multiplying and count how many 2's you need

Answer 17

its 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256 so because I know its multipled by 2, 8 times, where do I go from there?

Answer 18

count how many 2's you used

Answer 19

8

Answer 20

8, so that is the answer
year 8 is when you get 256 runs

Answer 21

in other words
\[ 256= 2^8 \]
and log base 2 of 2^8 means "tell me what the exponent is"
the exponent is 8

Answer 22

if you need to use a calculator (which does not know base 2)
you can use base 10 (for example)
type log(256)
that gives you the exponent for 10 (which is not 2)
then divide that number by log(2)
you should get 8

Answer 23

thank you so much