SOLUTION: A function y is of the form y = ax^n + bx where a, b and n are real numbers. When x is equal to 1, 3 and 9 respectively, the corresponding values of y are 4, 6, and 15. Find two

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: A function y is of the form y = ax^n + bx where a, b and n are real numbers. When x is equal to 1, 3 and 9 respectively, the corresponding values of y are 4, 6, and 15. Find two      Log On


   

Question 976027: A function y is of the form y = ax^n + bx where a, b and n are real numbers. When x is equal to 1, 3 and 9 respectively, the corresponding values of y are 4, 6, and 15.
Find two (2) relations between a and b, not involving n. Hence, or otherwise, find the exact numerical values of a and b, and the value of n correct to 3 decimal places.
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
y+=+ax%5En+%2B+bx
.
.
.
When x=1,
4+=+a+%2B+b
1. a=4-b
.
.
.
When x=3,
2.6=a%2A3%5En%2B3b
.
.
.
When x=9,
15=a%2A9%5En%2B9b
3.15=a%2A3%5E%282n%29%2B9b
.
.
.
From eq. 2,
a%2A3%5En=3b-6
a%5E2%2A3%5E%282n%29=9%28b-2%29%5E2
From eq. 3,
a%2A3%5E%282n%29%2B9b=15
So,
a%5E2%2A3%5E%282n%29%2B9ab=15a
Comparing,
9%28b-2%29%5E2=15a-9ab
9%28b-2%29%5E2=3a%285-3b%29
So then substituting from eq. 1,
9%28b-2%29%5E2=3%284-b%29%285-3b%29
9%28b%5E2-4b%2B4%29=3%2820-12b-5b%2B3b%5E2%29
9b%5E2-36b%2B36=60-51b%2B9b%5E2
15b-24=0
15b=24
b=24%2F15
highlight%28b=8%2F5%29
Then,
a=4-8%2F5
a=20%2F5-8%2F5
highlight%28a=12%2F5%29
And finally,
6=%2812%2F5%29%2A3%5En%2B3%288%2F5%29
30=12%2A3%5En%2B24
6=12%2A3%5En
3%5En=1%2F2
n=log%283%2C%281%2F2%29%29
n=log%28%281%2F2%29%29%2Flog%28%283%29%29
highlight%28n=-log%28%282%29%29%2Flog%28%283%29%29%29