College Algebra, Chapter 5, 5.3, Section 5.3, Problem 70

Based from some Biologists, the equation $\log S = \log c + k \log A$ (The species-area relationship) is the model of the number of species $S$ in a fixed area $A$, where $c$ and $k$ are positive constants that depend on the type of species and habitat.

a.) Solve the equation for $S$


$
\begin{equation}
\begin{aligned}

\log S =& \log c + \log A^k
&& \text{Law of Logarithm } \log_a (A^C) = C \log_a A
\\
\\
\log S =& \log (cA^k)
&& \text{Law of Logarithm } \log_a (AB) = \log_a A + \log_a B
\\
\\
S =& cA^k
&& \text{Because $\log$ is one-to-one}

\end{aligned}
\end{equation}
$



b.) Show that if $k = 3$, then doubling the area increases the number of species eight fold, using part (a).


$
\begin{equation}
\begin{aligned}

S =& c(2A)^3
\qquad \text{Substitute } k = 3
\\
\\
S =& 8cA^3

\end{aligned}
\end{equation}
$


It shows that when the area is doubled the number of species increases eight times.