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Section 1.4 Logarithmic Functions

Watermark text: DRAFT

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 \(f\text{.}\)

This is a picture of a logarithmic function. The two points shown are (1, 0) and (9, 2).
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 the logarithmic statement into an exponential one)}\\ \amp \implies a=3 \amp\amp \text{(take the positive square root of } 9 \text{ because bases of logs 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.

Exercises Exercises

1.

The graph of \(f(x)=\log_a(x)\) is given below. Find the value of \(a\text{.}\) Note, the points \(\left( 1,0 \right)\) and \(\left( 25,4 \right)\) are on the graph of \(f\text{.}\)

This is a picture of a logarithmic function. The two points shown are (1, 0) and (25, 4).

Each table represents a table-of-values for a function \(f(x)=\log_a(x)\text{.}\) Find the value of \(a\text{.}\)

2.
\(x\) \(0.000125\) \(0.5\) \(1\) \(2\sqrt{5}\) \(400\)
\(f(x)\) \(-3\) \(-1\) \(0\) \(0.5\) \(2\)
3.
\(x\) \(\frac{1}{9}\) \(1\) \(3\) \(81\) \(243\)
\(f(x)\) \(-4\) \(0\) \(2\) \(8\) \(10\)