Let's analyze the problem and solve it step by step.

The problem provides two functions:

1. $f(x) = b^x$ in Table I.
2. $g(x) = \log_b(x)$ in Table II.

Both functions involve a base $b$, which is the same positive constant for both tables.

We need to fill in the missing values in the tables.

Step 1: Relating the two functions

• Exponential Function: $f(x) = b^x$.
• Logarithmic Function: $g(x) = \log_b(x)$, which is the inverse of the exponential function.

Thus, if $f(x) = b^x$, then $g(x) = \log_b(f(x)) = x$, and vice versa.

Step 2: Analyze the given data

We'll begin by trying to deduce the base $b$ using the known values from Table I and Table II.

Table I (for $f(x) = b^x$):

• $f(0.585) = 1.5$
• $f(1) = ?$ (missing value)
• $f(3) = 8$
• $f(3.322) = 10$

Table II (for $g(x) = \log_b(x)$):

• $g(1.5) = 0.585$
• $g(2) = 1$
• $g(2.322) = ?$ (missing value)
• $g(5) = 2$
• $g(3) = 3$

Step 3: Determine $b$

From Table I, we have:

• $f(3.322) = 10 \Rightarrow b^{3.322} = 10$. Taking the logarithm: $b = 10^{\frac{1}{3.322}} \approx 2.154$ So, the base $b \approx 2.154$.

Step 4: Fill in missing values

Now that we know $b \approx 2.154$, we can use it to find the missing values.

For Table I:

1. $f(1) = b^1 = 2.154^1 = 2.154$.

For Table II:

We know $g(x) = \log_b(x)$.

1. $g(2.322) = \log_{2.154}(2.322) \approx 1.252$ (rounded to 3 decimal places).

Final Tables:

Table I:

\hline x & f(x) = b^x \\ \hline 0.585 & 1.5 \\ 1 & 2.154 \\ 3 & 8 \\ 3.322 & 10 \\ \hline \end{array}$$ #### Table II: $$\begin{array}{|c|c|} \hline x & g(x) = \log_b(x) \\ \hline 1.5 & 0.585 \\ 2 & 1 \\ 2.322 & 1.252 \\ 5 & 2 \\ 3 & 3 \\ \hline \end{array}$$ Let me know if you would like further details on any step or calculation! --- ### Related Questions: 1. What is the inverse of the function $$f(x) = b^x$$? 2. How do you calculate logarithms using a change of base formula? 3. What is the relationship between exponential and logarithmic functions? 4. How would the values change if the base $$b$$ were different? 5. Can you graph $$f(x) = b^x$$ and $$g(x) = \log_b(x)$$ to visualize the relationship? ### Tip: Remember, logarithms are the inverse operations of exponentials!