To find the logarithmic function that represents the data in the table, we need to observe the relationship between the x values and the corresponding fx values.
The table gives us the following pairs:
- (1, 0)
- (3, 1)
- (9, 2)
From these points, we can notice that the function appears to be a logarithmic function of the form:
f(x) = a * log_b(x)
We will determine the values of a and b. A common logarithmic base is usually base 10, but we can choose another base if necessary. Let’s see if we can fit the points.
1. For the first point (1, 0):
f(1) = a * log_b(1) = 0
Since the logarithm of 1 is 0 for any base, this point is satisfied for any a.
2. For the second point (3, 1):
f(3) = a * log_b(3) = 1
This implies that a = 1 / log_b(3).
3. For the third point (9, 2):
f(9) = a * log_b(9) = 2
Using the properties of logarithms, log_b(9) = log_b(3^2) = 2 * log_b(3), leads to:
f(9) = a * 2 * log_b(3) = 2
Substituting a from the previous equation:
(1 / log_b(3)) * 2 * log_b(3) = 2
This confirms that our relationships hold.
From here, we can conclude a logarithmic function could be:
f(x) = log_b(x) for an appropriate base b.
To verify:
- f(1) = log_b(1) = 0
- f(3) = log_b(3) = 1
- f(9) = log_b(9) = 2
In this case, for base 3, our logarithmic function becomes:
f(x) = log_3(x).
This function accurately represents the pairs given in the data table.