Question

HELP! I don't understand this at all :( To find out how long it will take for $500 to double when invested at 5% annual interest compounded twice a year, you solve the following equation: 1000 = 500 (1 + ) 2t. Use complete sentence to describe each step of math needed to solve this equation.

Answer 1

what is in parenthesis?

Answer 2

(1+ .05/2)

Answer 3

1000 = 500 (1+(.05/2))^2t sorry i forgot the carrot!

Answer 4

the formula is
\[ 1000 = 500 (1+\frac{0.05}{2})^{2t} \]

Answer 5

where t is in years.

Answer 6

to solve, the first step is divide both sides by 500

Answer 7

\[ 2= 1.025^{2t} \]

Answer 8

how did you get 1.025^2t

Answer 9

I divided 2 into 0.05 to get 0.025 and added it to 1

Answer 10

one way to "bring down the exponent" 2t is use logarithms. Let's use log base 10:
log(2)= log(1.025^2t)
then use this very useful property
\[ log(a^b)= b \cdot log(a) \]

Answer 11

but i thought you were supposed to be dividing 500 into it

Answer 12

we start with
1000= 500 (1+0.05/2)^(2t)

divide both sides by 500 to get
2= (1+0.05/2)^(2t)

divide 0.05 by 2:
2= (1+0.025)^(2t)

add 1 + 0.025= 1.025

2= (1.025)^(2t) or simply 2= 1.025^(2t)
follow?

Answer 13

oh nevermind haha ignore what i jsut said

Answer 14

i follow

Answer 15

take the log (base 10 tho it does not matter which base we use) of both sides
log(2)= log(1.025^(2t))

can you use the rule
\[ log(a^b)= b \cdot log(a) \]
to rewrite this equation?

Answer 16

i dont know what base ten means... but i have no idea how to rewrite it like that

Answer 17

would it be like log(2^10) = 10 * log(1.025^(2t))

Answer 18

using the rule means match the rule to your expression.
if you have log(2^10) you can rewrite it as 10*log(2) (2 matches a and 10 matches b in the rule).

in this case
match the rule to log(1.025^(2t))

Answer 19

10*log(2) = 10*log(2)

Answer 20

theyre the same

Answer 21

in this case
match the rule to log(1.025^(2t))

Answer 22

forget 10*log(2) that was an example. look at

log(1.025^(2t))
which part matches the a in the rule. which part matches the b in the rule

Answer 23

the 1.025 part is a

Answer 24

yes

Answer 25

2t is b

Answer 26

yes. so now use the rule to rewrite

log(1.025^(2t))

Answer 27

soooo 2t*log(1.025)

Answer 28

yes. so back to the problem
log(2)= log(1.025^(2t))
which is also (using the rule)
log(2)= (2t)*log(1.025)

now divide both sides by 2*log(1.025)

Answer 29

like this
\[ \frac{log(2)}{2\cdot log(1.025)}= \frac{t\cdot 2 \cdot log(1.025)}{2\cdot log(1.025)}\]

Answer 30

\[ \frac{log(2)}{2\cdot log(1.025)}= \frac{t\cdot \cancel{2 \cdot log(1.025)}}{\cancel{2\cdot log(1.025)}} \]