Need help finding out what the variable x is equal to in this exponential decay eq.

(√5 / 25) = 5^{3-x}

How do you find the value of x? My calculator says that x is equal to 4.5. However, can someone please show me the work to finding the answer to this equation? Thanks!

Re: Need help finding out what the variable x is equal to in this exponential decay e

Hey sameeranand21.

Are you familiar with logarithms in arbitrary bases?

SQRT(5)/25 = SQRT(5)/[SQRT(5)*5*SQRT(5)] = 1/[5*SQRT(5)] = 5^(-3/2) = 5^(3-x).

So you can use logarithms or use the fact that if 5^x = 5^y then x = y (also known as injective property of function).

Re: Need help finding out what the variable x is equal to in this exponential decay e

I am not understanding how you found this: SQRT(5)/25 = SQRT(5)/[SQRT(5)*5*SQRT(5)]

Nor am I understanding how this 1/[5*SQRT(5)] equals this 5^(-3/2).

Re: Need help finding out what the variable x is equal to in this exponential decay e

1/a^x = a^(-x) and a^(x)*a^(y) = a^(x+y).

SQRT(5) = 5^(1/2) 5*SQRT(5) = 5^(1+1/2) = 5^(3/2)

1/5^(3/2) = 5^(-3/2)