The question involves finding a function that represents the linearized data. The table in the image shows values for $x$ and the logarithmic values of $f(x)$ in base 5. Based on the linearized form of the function, the function will typically take the form $f(x) = A \cdot B^x$, where $A$ is a constant and $B$ is the base of the exponential.

Let’s analyze the values from the table:

• For $x = 0$, $\log_5(f(x)) = 8$
• For $x = 1$, $\log_5(f(x)) = 9$
• For $x = 2$, $\log_5(f(x)) = 10$
• For $x = 3$, $\log_5(f(x)) = 11$
• For $x = 4$, $\log_5(f(x)) = 12$

Notice that the logarithmic values are increasing by 1 for each increment of $x$. This suggests that the function $f(x)$ follows an exponential growth pattern. Based on the data, we can now express this in exponential form.

The general form of the function will be: $f(x) = A \cdot 5^x$ where $A$ is the constant that we can solve for using the given data.

At $x = 0$, the value of $\log_5(f(x)) = 8$, which means: $\log_5(f(0)) = 8 \quad \Rightarrow \quad f(0) = 5^8 = 390,625$

Therefore, the function is: $f(x) = 390,625 \cdot 5^x$

The correct answer is the third option: $f(x) = 390,625 \cdot 5^x$

Would you like more details on how to approach this problem or clarification on any steps? Feel free to ask!

Here are 5 related questions to expand on the topic:

1. How can we apply logarithmic functions to solve exponential equations?
2. What is the relationship between a logarithmic and exponential function?
3. How do you determine the base of an exponential function?
4. How do we linearize data for exponential functions in real-life scenarios?
5. How does the constant $A$ affect the growth rate of the exponential function?

Tip: When dealing with logarithmic functions, remember that the base of the logarithm is crucial in determining the function's growth behavior.