We haven't received confirmation from hannoonour that variable n represents days, so let's assume that it does. (Hopefully, hannoonour was able to finish their exercise.)
Here's a worked
example, for future readers.
The given exponential-decay formula is:
\(\displaystyle \quad M = I\;(0.5^{n})\)
where the parameter [imath]I[/imath] = initial tungsten-187 (in grams)
the independent variable [imath]n[/imath] = elapsed time (in days)
the dependent variable [imath]M[/imath] = remaining tungsten-187 (in grams)
How much time is needed for 749 grams of tungsten-187 to decay to 16 grams?
Substitute the values given for [imath]M[/imath] and [imath]I[/imath]:
\(\displaystyle \quad 16 = 749 \cdot 0.5^{n}\)
Divide each side by 749, to isolate the exponential part:
\(\displaystyle \quad\frac{16}{749} = 0.5^{n}\)
Take the natural logarithm of each side:
\(\displaystyle \quad ln\bigg(\frac{16}{749}\bigg) = ln\bigg(0.5^{n}\bigg)\)
Move the time variable out of the exponent position, by applying this property of logarithms:
ln(cx) = x∙ln(c)
\(\displaystyle \quad ln\bigg(\frac{16}{749}\bigg) = n \cdot ln(0.5)\)
Divide each side by [imath]ln(0.5)[/imath], to solve for n:
\(\displaystyle \quad \frac{ln\big(\frac{16}{749}\big)}{ln(0.5)} = n\)
Use a scientific calculator — or google ln(16/749)/ln(0.5) — to evaluate the left-hand side:
\(\displaystyle \quad 5.5488 = n \quad\)
(rounded to four decimal places)
It takes a little more than 5½ days, for 749 grams of tungsten-187 to decay to 16 grams.
We could also convert the fractional-day part (0.5488) to hours and minutes, if required:
n ≈ 5 days 13 hours 10¼ minutes
Let's check the answer:
M = (749)(0.5)^5.5488
M = (749)(0.021362)
M = 16.0001
[imath]\;[/imath]
