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