Hey guys. I'm trying to figure out the amount of time it would take for a tungsten isotope to decay from 64g to 4g if the decay rate is .5/day.
What I have so far is:
4=64(1/2)x
4=64(2-1)x
4=64(2-x)
I can figure out the answer is 4 because it's an easy decay rate, but what I can't figure out is how to show it. Maybe someone has an easy trick for decay rates? For some reason this kicking my butt.
Have you studied logarithms at all yet? If not, then that's probably why they gave you this particular form of exponential equation, because
it's the only sort you can do without logs; that is, it's one where you can restate things in terms of one base, and then solve by comparing the powers.
In your particular case (and this is how you're probably expected to do this), you have the following steps:
. . . . .\(\displaystyle 4\, =\, 2^2\)
. . . . .\(\displaystyle 64\, =\, 8^2\, =\, \left(2^3\right)^2\, =\, 2^{(3\,\cdot\,2)}\, =\, 2^6\)
. . . . .\(\displaystyle \left(\dfrac{1}{2}\right)^x\, =\, \left(2^{-1}\right)^x\, =\, 2^{-x}\)
. . . . .\(\displaystyle (64)\left(2^{-x}\right)\, =\, \left(2^6\right)\left(2^{-x}\right)\, =\, 2^{(6\, -\, x)}\)
Now that both sides have the same base of 2, we can "equate the powers"; that is, since the bottoms are the same and since the two sides are the same (because that's what the "equals" sign in the middle
means), then logically the powers also have to be the same. So then we can create a new equation:
. . . . .\(\displaystyle 2\, =\, 6\, -\, x\)
...which is an equation of the sort that we
can solve without logs. And, of course, one quickly arrives at the solution you'd already found; namely, that x must be equal to 4.
