What is the logarithmic function modeled by the following table?
x f(x)
9 2
27 3
81 4
To determine the logarithmic function modeled by the given table, we need to understand the relationship between the values of x and f(x). In a logarithmic function, the logarithm of a number is the exponent to which another fixed value, called the base, must be raised to produce that number.
Let's take a closer look at the table:
x f(x)
9 2
27 3
81 4
From the table, notice that each value of x is a power of 3 (3^2 = 9, 3^3 = 27, and 3^4 = 81). This suggests that the base of the logarithmic function is likely 3.
To find the logarithm, we can write each equation in logarithmic form:
log₃(9) = 2
log₃(27) = 3
log₃(81) = 4
The general form of a logarithmic function is y = logₐ(x), where a is the base and x is the argument. Therefore, we can conclude that the logarithmic function modeled by the table is:
f(x) = log₃(x)
To find the logarithmic function modeled by the table, we need to examine the relationship between x and f(x). In this case, it looks like f(x) is the exponent to which 3 should be raised to equal x.
Let's rewrite the table using logarithmic notation:
x f(x)
9 log₃(9)
27 log₃(27)
81 log₃(81)
Now, let's evaluate these logarithms:
x f(x)
9 log₃(9) = 2
27 log₃(27) = 3
81 log₃(81) = 4
From the table, we can see that log₃(x) = f(x). So, the logarithmic function modeled by the table is f(x) = log₃(x).