Complete the given table.
Exponential Form

Logarithmic Form
log2^(2) = 1
log4

1
256

= −4
log4^(16) = 2
log10^(100000) = 5
log3^(81) = 4
log4^(1) = 0

Let's break down and complete the table by converting between exponential and logarithmic forms.

The given logarithmic expressions in the form \( \log_b(a) = c \) can be rewritten into exponential form using the rule:

\[
\log_b(a) = c \quad \text{means} \quad b^c = a
\]

### Here’s how it works:

1. **\( \log_2(2) = 1 \)**:
   - **Exponential form**: \( 2^1 = 2 \)

2. **\( \log_4(x) = -4 \)**:
   - We convert it to exponential form: 
     \[ 4^{-4} = x \]
     Solving, we get \( x = \frac{1}{4^4} = \frac{1}{256} \).
   - **Exponential form**: \( 4^{-4} = \frac{1}{256} \)

3. **\( \log_4(16) = 2 \)**:
   - **Exponential form**: \( 4^2 = 16 \)

4. **\( \log_{10}(100000) = 5 \)**:
   - **Exponential form**: \( 10^5 = 100000 \)

5. **\( \log_3(81) = 4 \)**:
   - **Exponential form**: \( 3^4 = 81 \)

6. **\( \log_4(1) = 0 \)**:
   - **Exponential form**: \( 4^0 = 1 \)

### Completed Table:

| Logarithmic Form               | Exponential Form    |
|---------------------------------|---------------------|
| \( \log_2(2) = 1 \)             | \( 2^1 = 2 \)       |
| \( \log_4(x) = -4 \)            | \( 4^{-4} = \frac{1}{256} \) |
| \( \log_4(16) = 2 \)            | \( 4^2 = 16 \)      |
| \( \log_{10}(100000) = 5 \)     | \( 10^5 = 100000 \) |
| \( \log_3(81) = 4 \)            | \( 3^4 = 81 \)      |
| \( \log_4(1) = 0 \)             | \( 4^0 = 1 \)       |

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Would you like a further breakdown of any of these conversions or more details? Here are some related questions:

1. What is the general relationship between logarithms and exponents?
2. How do you solve for unknowns in logarithmic equations?
3. How can logarithmic properties simplify complex equations?
4. Can logarithms be used to solve exponential growth problems?
5. What are common mistakes when converting between logarithmic and exponential forms?

**Tip**: Always remember that \( \log_b(1) = 0 \) for any base \( b \), since \( b^0 = 1 \).

This problem demonstrates how to complete a table by converting logarithmic forms to exponential forms. It covers key concepts of logarithms and exponents, suitable for students in Grades 9-12. Learn how to rewrite logarithmic expressions like log2(2) = 1, log4(16) = 2, and more.

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