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The tables in the image ask you to fill in values based on two functions:

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

### Understanding the Tables:
- In **Table I**, we are given the values of \( x \) and are asked to find the values of \( f(x) = b^x \) for specific \( x \) values.
- In **Table II**, we are given the values of \( x \) and need to find \( g(x) = \log_b(x) \), the logarithm of \( x \) with base \( b \).

Both tables involve the same constant base \( b \).

### Step 1: Calculate the Base \( b \)
Looking at **Table I**, we have the following data points:
- When \( x = 1 \), \( f(x) = b^1 = 2 \). Therefore, \( b = 2 \).

Now, we can use \( b = 2 \) to fill in the missing values in both tables.

### Step 2: Fill in Table I (Using \( f(x) = 2^x \))
- For \( x = 0.1 \): 
  \[
  f(0.1) = 2^{0.1} \approx 1.072
  \]
- For \( x = 2.041 \): 
  \[
  f(2.041) = 2^{2.041} \approx 4.126
  \]

Thus, the completed **Table I** is:
\[
\begin{array}{|c|c|}
\hline
x & f(x) = 2^x \\
\hline
0.1 & 1.072 \\
1 & 2 \\
2.041 & 4.126 \\
\hline
\end{array}
\]

### Step 3: Fill in Table II (Using \( g(x) = \log_2(x) \))
- For \( x = 0.1 \):
  \[
  g(0.1) = \log_2(0.1) \approx -3.322
  \]
- For \( x = 2 \):
  \[
  g(2) = \log_2(2) = 1
  \]
- For \( x = 100 \):
  \[
  g(100) = \log_2(100) \approx 6.644
  \]

Thus, the completed **Table II** is:
\[
\begin{array}{|c|c|}
\hline
x & g(x) = \log_2(x) \\
\hline
0.1 & -3.322 \\
1 & 0 \\
2 & 1 \\
100 & 6.644 \\
\hline
\end{array}
\]

### Final Answer:
- Table I: \( f(0.1) \approx 1.072 \), \( f(2.041) \approx 4.126 \)
- Table II: \( g(0.1) \approx -3.322 \), \( g(100) \approx 6.644 \)

Let me know if you need further details or explanations!

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Here are 5 related questions you might find helpful:
1. What is the formula for converting logarithms from one base to another?
2. How do you calculate logarithms with non-integer results by hand?
3. What are the properties of exponential functions with different bases?
4. How does changing the base of an exponential function affect its growth rate?
5. How can we use logarithms to solve exponential equations?

**Tip**: Remember, for any base \( b \), \( \log_b(b) = 1 \) and \( \log_b(1) = 0 \).

Preencha os valores que estão faltando nas tabelas.

Explore the completion of tables for the exponential function b^x and the logarithmic function log_b(x) with a base of 2. This guide covers the calculations for missing values in the tables, including detailed steps and formulas. Suitable for high school students studying advanced functions.

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