Let’s solve the problem step by step.

Problem Description

• Table I provides $f(x) = b^x$ for certain $x$-values.
• Table II provides $g(x) = \log_b(x)$ for the same constant $b$.
• The task is to fill in the missing values in both tables, given $b$ is constant across functions.

Step 1: Find $b$

From Table I, we know:

• When $x = 1$, $f(x) = b^1 = b = 3$.

Thus, $b = 3$.

Step 2: Complete Table I

Using $f(x) = b^x = 3^x$:

• $x = 0.792$: $f(x) = 3^{0.792}$.
• $x = 1.661$: $f(x) = 3^{1.661}$.

Calculations:

1. $3^{0.792} \approx 2.207$ (rounded to three decimals).
2. $3^{1.661} \approx 6.741$ (rounded to three decimals).

Updated Table I:

Step 3: Complete Table II

Using $g(x) = \log_b(x) = \log_3(x)$:

• $x = 3$: $g(x) = \log_3(3) = 1$.
• $x = 10$: $g(x) = \log_3(10)$.
• $x = 0.792$: $g(x) = \log_3(0.792)$.

Calculations:

1. $\log_3(0.792) = \frac{\log(0.792)}{\log(3)} \approx -0.129$ (rounded to three decimals).
2. $\log_3(10) = \frac{\log(10)}{\log(3)} \approx 2.096$ (rounded to three decimals).

Updated Table II:

Final Answer:

Table I:

Table II:

Let me know if you’d like me to explain any part in detail!

Relative Questions

1. What is the relationship between $f(x)$ and $g(x)$ in exponential and logarithmic functions?
2. How can we use logarithms of different bases for similar calculations?
3. How does the constant $b$ affect the growth of $f(x)$?
4. What are the real-world applications of exponential and logarithmic functions?
5. How can we verify our calculations without using a calculator?

Tip:

Always check your calculations by substituting values back into the original equations!