Logarithms?

Question

The demand for a new DVD player depends on its price--lower the price and the demand increases. The demand equation for the player is found to be P=0.5(10)^(0.002x). wher eP is the price and x is the demand. Find the difference in demand when the player is priced at $350 and $450


---I was doing fine with logarithms untill it came to equations like this one. No matter what I do Im not getting the correct answers. PLEASE HELP :) showing all the work so I can learn to do these correctly. Thanks so much.

Answer 1

Well first sustitute in the numbers, ie.

350=0.5(10)^(0.002x)
700=10^(0.002x)
log(700)= 0.002x log(10)
2.845 = 0.002x
x = 1422.5
and
450=0.5(10)^(0.002x)
900=10^(0.002x)
log(900) = 0.002x log(10)
2.954 = 0.002x
x = 1477.1

Therefore the difference in demand is 1477.1 - 1422.5 = 54.6

Answer 2

P=0.5(10)^(0.002x)
log(P) = (.002x)*log(0.5*10)
x = (1/0.002)log(P)/log(5)

when P = 350...
x = 1819.87

when P = 450...
x = 1897.94

Difference in x = 78.07

Answer 3

P = 0.5 x 10^(0.002 x)
2P = 10^(0.002 x)
log 2P = 0.002 x (all logs to base ten)

x1 = log 2P / 0.002
x1 = log 700 / 0.002
x1 = 1423

x2 = log 900 / 0.002
x2 = 1477

Difference in demand = 54

Answer 4

Solve for x; then plug in the two values of P and take the difference in x:

P=0.5*(10)^(0.002x)

LN P=LN 0.5*(10)^(0.002x) =
LN 0.5 + LN (10)^(0.002x) =
LN 0.5 + (.002x)*LN (10).

LN P = LN 0.5 + (.002x)*LN (10)
(LN P - LN 0.5)/(.002*LN 10) = x

(LN 350 - LN 0.5)/(.002*LN 10) =1422.55


(LN 450 - LN 0.5)/(.002*LN 10) = 1477.12

difference = 54.57