Consider the equation

Question

 

 

(a) By using the substitution u = 3^x show that the equation 9^x −3(3^x) = −2 can be written as u² −3u + 2 = 0.

 

(b) Find the value(s) of u which satisfy the equation in part (b) and hence ?nd the value(s) of x which satisfy the equation 9^x −3(3^x) = −2.

Answer 1

9^x - 3(3^x) = -2

9^x - 3(3^x) + 2 = 0

(3²)^x - 3(3^x) + 2 = 0

3^2x - 3(3^x) + 2 = 0

(3^x)² - 3(3^x) + 2 = 0

If u = 3^x

u² - 3u + 2 = 0

Answer 2

a) u=3^x

 

u²=(3^x)²=9^x

 

u² −3u + 2=0 the solutions are {1,2}

 

3^x=1 will get a values of 0 to x

 

3^x=2

Taking log on both sided.(base 10

 

x.log(3)=log(2)

 

x=log(2)/log(3)= 0.63093

 

3 ^0.63093=2(verification)