https://file.aiquickdraw.com/mathbot/file/1727325439460bv767a4l.jpg

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:
\[
\begin{array}{|c|c|}
\hline
x & f(x) = b^x \\
\hline
0.585 & 1.5 \\
1 & 2.154 \\
3 & 8 \\
3.322 & 10 \\
\hline
\end{array}
\]

#### 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!

Learn how to solve missing values in a table involving exponential functions f(x) = b^x and logarithmic functions g(x) = log_b(x). This problem shows how to find the base b and fill in missing values. Suitable for high school students studying logarithmic and exponential relationships.

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