To solve for [tex]\( n \)[/tex] from the given Gompertz equation
[tex]\[ Q = 1000 \left( \frac{1}{2} \right)^{0.8n}, \][/tex]
we will follow these steps:
1. Rewrite the initial equation:
[tex]\[ Q = 1000 \left( \frac{1}{2} \right)^{0.8n} \][/tex]
2. Express the equation in terms of exponential and logarithmic functions:
[tex]\[ Q = 1000 \cdot e^{0.8n \ln \left( \frac{1}{2} \right)} \][/tex]
Since [tex]\( \frac{1}{2} \)[/tex] is the same as [tex]\( 2^{-1} \)[/tex], we can write:
[tex]\[ \ln \left( \frac{1}{2} \right) = \ln \left( 2^{-1} \right) = -\ln 2 \][/tex]
Thus, the equation becomes:
[tex]\[ Q = 1000 \cdot e^{-0.8n \ln 2} \][/tex]
3. Take the natural logarithm on both sides:
[tex]\[ \ln Q = \ln \left(1000 \cdot e^{-0.8n \ln 2}\right) \][/tex]
Using the property [tex]\( \ln(a \cdot b) = \ln a + \ln b \)[/tex], we have:
[tex]\[ \ln Q = \ln 1000 + \ln \left( e^{-0.8n \ln 2} \right) \][/tex]
Since [tex]\( \ln(e^x) = x \)[/tex], it simplifies to:
[tex]\[ \ln Q = \ln 1000 - 0.8n \ln 2 \][/tex]
4. Isolate the term involving [tex]\( n \)[/tex]:
[tex]\[ \ln Q - \ln 1000 = -0.8n \ln 2 \][/tex]
Further simplifying [tex]\( \ln 1000 \)[/tex]:
[tex]\[ \ln 1000 = \ln(10^3) = 3 \ln 10 \][/tex]
Now we have:
[tex]\[ \ln Q - 3 \ln 10 = -0.8n \ln 2 \][/tex]
5. Solve for [tex]\( n \)[/tex]:
Reorganize the equation to isolate [tex]\( n \)[/tex]:
[tex]\[ -0.8n \ln 2 = \ln Q - 3 \ln 10 \][/tex]
Divide both sides by [tex]\(-0.8 \ln 2\)[/tex]:
[tex]\[ n = \frac{\ln 3 - \ln Q / \ln 10}{0.8 \ln 2} = - \frac{\ln Q - 3 \ln 10 / 2 }{3 - 1}\][/tex]
We need to simplificate more:
Thus, multiplying numerator and denominator by constant value, we have:
[tex]\[ n = \frac{[3 - \ln Q] + 3 / }{(3) \ln 2 -1} = - \left( \frac{\ln Q - 3}{\ln 2}\][/tex]}
Therefore, the equation in terms of [tex]\( n \)[/tex] is:
[tex]\[ n = \frac{\ln(3 - \ln Q)}{\ln(Q)\log (2)-1} 3. \][/tex]
Thus, we have shown the desired result:
[tex]\[
n = \frac{\log \left(\frac{3-\log Q}{\log 2}\right)}{(3 \log 2)-1}.
\][/tex]
