Question

If 9^x = 9^x+2 - 240 find the values of x​

Answer 1

Step-by-step explanation:

Given :-

9^x = 9^(x+2) - 240

To find :-

Find the value of x ?

Solution :-

Given equation is 9^x = 9^(x+2) - 240

=> 9^x -9^(x+2) = - 240

=> 9^(x+2) - 9^x = 240

=> (9^x×9²) -9^x = 240

Since a^m × a^n = a^(m+n)

=> 9²×9^x - 9^x = 240

=> 81 × 9^x - 9^x = 240

=> (81-1)×9^x = 240

=> 80×9^x = 240

=> 9^x = 240/80

=> 9^x = 3

=> (3²)^x = 3

=> 3^(2×x) = 3

Since (a^m)^n = a^mn

=> 3^2x = 3

=> 3^2x = 3¹

If bases are equal then exponents must be equal.

=> 2x = 1

=>x = 1/2

Therefore, x = 1/2

Answer:-

The value of x for the given problem is 1/2

Check :-

If x = 1/2 then

LHS = 9^x = 9^1/2 = 3^2/2 = 3

RHS = 9^x+2 - 240

=> 9^(1/2+2)-240

=> 9^(1+4)/2 -240

=> 9^(5/2) -240

=> (3²)^5/2 -240

=> 3^10/2 -240

=> 3^5 - 240

=> 243-240

=> 3

LHS = RHS is true for x = 1/2

Used formulae:-

• a^m × a^n = a^(m+n)

• (a^m)^n = a^mn

• If bases are equal then exponents must be equal.