8. In a controlled experiment, the number of microbes, \(N\), present in a culture \(T\) days after the start of the experiment were counted.
\(N\) and \(T\) are expected to satisfy a relationship of the form
$$N = a T ^ { b } \quad \text { where } a \text { and } b \text { are constants }$$
- Show that this relationship can be expressed in the form
$$\log _ { 10 } N = m \log _ { 10 } T + c$$
giving \(m\) and \(c\) in terms of the constants \(a\) and/or \(b\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d8e25332-3a45-43ca-a5b8-0a16f47f13b9-24_1223_1043_895_461}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows the line of best fit for values of \(\log _ { 10 } N\) plotted against values of \(\log _ { 10 } T\) - Use the information provided to estimate the number of microbes present in the culture 3 days after the start of the experiment.
- With reference to the model, interpret the value of the constant \(a\).