# MTH 111–112 Supplement

## Section1.4Logarithmic Functions

### Example1.4.1.

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.

### ExercisesExercises

#### 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 $$f\text{.}$$

#### Find the Base.

In Exercises 2–3, each table represents a table-of-values for a function $$f(x)=\log_a(x)\text{.}$$ Find the value of $$a\text{.}$$
##### 2.
 $$x$$ $$0.000125$$ $$0.05$$ $$1$$ $$2\sqrt{5}$$ $$400$$ $$f(x)$$ $$-3$$ $$-1$$ $$0$$ $$0.5$$ $$2$$
##### 3.
 $$x$$ $$\frac{1}{9}$$ $$1$$ $$3$$ $$81$$ $$243$$ $$f(x)$$ $$-4$$ $$0$$ $$2$$ $$8$$ $$10$$