If 9^x+2=240+9^x, then the value of x is:

1957 AMC 12 Problem 33

Attempt Problem 33 of the 1957 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1957 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

33.

9^{x+2}=240+9^x,

then the value of

x

is:

0.1

0.2

0.3

0.4

0.5

Rewrite

9^{x+2}

81\cdot9^x

After collecting like terms, express

9^x

as a power of

3

Factoring

9^x,

81\cdot9^x-9^x=240,

80\cdot9^x=240

and

9^x=3.

Since

9^{\frac{1}{2}}=3,

we have

x=\frac{1}{2}=0.5.

Therefore, the correct answer is E.