if M = 3 x 6 ^ (-t)
find the half life of M
To find the half-life of M in the equation M = 3 x 6^(-t), we need to solve for t when M is equal to half its initial value.
The initial value of M can be calculated by setting t = 0:
M(0) = 3 x 6^0 = 3 x 1 = 3
To find the value of t when M is equal to half its initial value (M/2 = 3/2), we substitute M/2 into the equation:
M/2 = 3 x 6^(-t)
Next, we can solve this equation for t. To do this, we'll isolate the exponential term:
6^(-t) = (M/2) / 3
Now, take the logarithm of both sides of the equation. Any base can be used, but choosing the natural logarithm (ln) is common:
ln(6^(-t)) = ln((M/2) / 3)
Using the property of logarithms, we can bring the exponent down:
(-t)ln(6) = ln((M/2) / 3)
To solve for t, we need to isolate the variable. Divide both sides by (-ln(6)):
t = -ln((M/2) / 3) / ln(6)
Now, plug in the values for M and solve for t:
t = -ln((3/2) / 3) / ln(6)
Using a calculator, we can evaluate this expression to find the value of t.