Example 1.4.1.
The graph of \(f(x)=\log_a(x)\) is given in the graph below. Find the value of \(a\text{.}\) Note, the points \(\left( 1,0 \right)\) and \(\left( 9,2 \right)\) are on the graph of \(f\text{.}\)
Solution.
Since the function has the form \(f(x)=\log_a(x)\) and \(\left( 9,2 \right)\) is on the graph, we know that \(f(9)=2\text{.}\) Thus,
\begin{align*}
f(9)=2 \amp \implies \log_a(9)=2 \amp\amp \text{(since } f(9)=\log_a(9) \text{)}\\
\amp \implies a^2=9 \amp\amp \text{(translate to an exponential statement)}\\
\amp \implies a=3 \amp\amp \text{(positive square root because bases are positive)}
\end{align*}
Notice that we didn’t attempt to use \(\left( 1,0 \right)\text{,}\) the other obvious point on the graph of \(f(x)=\log_a(x)\text{,}\) to find the value of \(a\text{.}\) Why not? The point \(\left( 1,0 \right)\) is on the graph of all functions of the form \(f(x)=\log_a(x)\text{,}\) so it doesn’t provide information that will help us find the paerticular function graphed here.
