SOLUTION: A power function with form y= a x^p passes through the points (3, 2/81) and (6, 1/1296). Find the constants a and p I found a=6 please tell me the value of p?

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: A power function with form y= a x^p passes through the points (3, 2/81) and (6, 1/1296). Find the constants a and p I found a=6 please tell me the value of p?      Log On


   

Question 975062: A power function with form y= a x^p passes through the points (3, 2/81) and (6, 1/1296). Find the constants a and p
I found a=6 please tell me the value of p?
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

y=+a%2Ax%5Ep passes through the points (3, 2%2F81) and (6, 1%2F1296)
Find the constants a and p.


y=+a%2Ax%5Ep
log%28y%29=+log%28a%2Ax%5Ep+%29
log%28y%29=+log%28a%29%2Blog%28+x%5Ep+%29
log%28y%29=+log%28a%29%2Bp%2Alog%28+x+%29
if given the points :
(3,+2%2F81)=> x=3 and y=2%2F81
and
(6, 1%2F1296)=> x=6 and y=1%2F1296
so, we use them to form a system of equations

log%28y%29=+log%28a%29%2Bp%2Alog%28+x+%29 substitute x=3 and y=2%2F81
log%282%2F81%29=+log%28a%29%2Bp%2Alog%28+3+%29 .....eq.1
log%28y%29=+log%28a%29%2Bp%2Alog%28+x+%29 substitute x=6 and y=1%2F1296
log%281%2F1296%29=+log%28a%29%2Bp%2Alog%28+6%29 .....eq.2


log%282%29-log%2881%29=+log%28a%29%2Bp%2Alog%28+3+%29 .....eq.1
log%281%29-log%281296%29=+log%28a%29%2Bp%2Alog%28+6%29+ .....eq.2..........write 81 as 3%5E4 and 1296 as 3%5E4%2A2%5E4
--------------------------------------------------------------
log%282%29-log%283%5E4%29=+log%28a%29%2Bp%2Alog%28+3+%29 .....eq.1
log%281%29-log%283%5E4%2A2%5E4%29=+log%28a%29%2Bp%2Alog%28+2%2A3%29 .....eq.2
--------------------------------------------------------------------
log%282%29-4log%283%29=+log%28a%29%2Bp%2Alog%28+3+%29 .....eq.1
.....eq.2...substitute log%281%29=0
--------------------------------------------------------------------------------
log%282%29-4log%283%29=+log%28a%29%2Bp%2Alog%28+3+%29 .....eq.1
0-4log%283%29-4log%282%29%29=+log%28a%29%2Bp%2A%28log%28+2%29%2Blog%283%29%29 .....eq.2
------------------------------------------------------------------subtract 1 from 2

-5log%282%29=+p%2A%28log%28+2%29%2Blog%283%29-p%2Alog%28+3+%29%29
-5log%282%29=+p%2Alog%28+2%29

p=-5
log%282%29-4log%283%29=+log%28a%29%2Bp%2Alog%28+3+%29 .....eq.1...substitute -5 for p
log%282%29-4log%283%29=+log%28a%29-5%2Alog%28+3+%29+
log%282%29-4log%283%29%2B5%2Alog%28+3+%29+=+log%28a%29+
log%282%29%2Blog%28+3+%29+=+log%28a%29+
log%282%2A3%29=+log%28a%29
log%286%29=+log%28a%29+
a=6
so, your function is: y=+6x%5E%28-5%29 => y=+6%281%2Fx%5E5%29 => y=+6%2Fx%5E5