About Indices

Question

1. 2e^x = 3+14e^-x ((find x))) 

2. Given that y=ax^b +2 and that y=9 when x=2 and y=37 when x=4, find the value of a and b 

Answer 1

(1)

 

2e^x = 3 + 14e^-x

 

Do a substitution:

 

y = e^x ⇒ e^-x = 1/y

 

2y = 3 + 14/y

 

2y² = 3y + 14

 

2y² - 3y - 14 = 0

 

Solve this quadratic equation for y, and then

 

x = ln(y)

 

However, the quadratic equation will give 2 solutions, one positive and one negative.  Since you do not take ln of a negative number, keep only the positive solution, and test the answer.

 

(2)

 

y = ax^b + 2

 

9 = a(2^b) + 2

37 = a(4^b) + 2

 

⇒

 

7 = a(2^b)

35 = a(4^b)

 

Divide bottom equation by top equation.

 

5 = 4^b/2^b = (2²)^b/2^b = 2^2b/2^b = 2^b

 

log5 = b log2

 

b = log5/log2 [true for any base]

 

7 = a(2^log5/log2) = a(5)

 

a = 7 / 2^log5/log2 = 7/5

 

y = (7/5)x^log5/log2 + 2

 

Test the answers.