Question

The exponential formula for the half-life of a radioactive isotope is

y=y0ekt,

where y is the amount of the isotope remaining after t years, y0 is the initial amount of the isotope, k is the decay constant, and e is the transcendental number approximately equal to 2.71828.

How can you rearrange the given formula to correctly find y0?

Answer 1

divide by e^(kt) ... y / [e^(kt)] = y0

Answer 2

let t = 0
e^0 = 1
so
y0 = y at t = 0
or if you know y and t
then
y0= y / e^kt = y e^-kt
HOWEVER , That is NOT the equation for the half life.
k by the way is a negative number as you wrote the equation
y = y0 e^kt
for half life
1/2 = y/y0 = e^kt
ln 0.5 = k t
-0.693 = k t
t = -.693 /k when it is half gone