Show that solving the equation 3^2x=4 by taking common logarithms of both sides is equivalent to solving it by taking logarithms of base 3 of both sides.

Step-by-step explanation:

Case I:  use common logs:

 2x log 3 = log 4, or 2x(0.47712) = 0.60206

 Solving for x, we get 0.95424x = 0.60206, and then x = 0.60206/0.95424.

 x is then x = 0.631

Case II:  use logs to the base 3:

 2x (log to the base 3 of 3) = (log to the base 3 of 4)

 This simplifies to 2x(1) = 2x = (log 4)/log 3 = 1.262.  Finally, we divide this

 result by 2, obtaining x = 0.631

answer: the answer is 35%

delaney would like to make a 5lb nut mixture that is 60% peanuts and 40% almonds.

she has several pounds of peanuts and several pounds of a mixture that is 20% peanuts and 80% almonds.

let 'p' represent the number of pounds of peanuts needed to make the new mixture, and  

let 'm' represent the number of pounds of the 80% almond-20% peanut mixture.

:

p + m = 5

therefore

m = (5-p)

write an equation based on the percent of peanuts

1p + .2m = .6(5)

replace m with (5-p)

p + .2(5-p) = .6(5)

p + 1 - .2p = 3

p - .2p = 3 - 1

.8p = 2

p = 2/.8

p = 2.5 lbs of peanuts required

then

5 - 2.5 = 2.5 lbs of the peanut/almond mixture required

:

a) what is the system that models this situation?

p + m = 5

1p + .2m = .6(5)