The given expression is \(\log_r(z) = m\), which is a logarithmic equation. To understand and solve this equation, we can convert it to its exponential form. 

### Exponential Form
The logarithmic equation \(\log_r(z) = m\) can be rewritten in exponential form as:

\[ z = r^m \]

where:
- \(r\) is the base of the logarithm,
- \(z\) is the argument of the logarithm,
- \(m\) is the result of the logarithm.

### Example Problem:
Let's solve for \(z\) if \(r = 2\) and \(m = 3\).

1. Given: \(\log_2(z) = 3\)
2. Convert to exponential form: \( z = 2^3 \)
3. Compute \( z \):

\[ z = 2^3 = 8 \]

### Solution:
\( z = 8 \)

### General Explanation:
1. **Conversion to Exponential Form**: \(\log_r(z) = m\) means \(z\) is \(r\) raised to the power of \(m\).
2. **Solving for Different Variables**:
    - If you know \(r\) and \(m\), you can find \(z\) by computing \(r^m\).
    - If you know \(z\) and \(m\), you can find \(r\) by taking the \(m\)-th root of \(z\), \(r = z^{1/m}\).
    - If you know \(r\) and \(z\), you can find \(m\) by using the logarithmic function, \(m = \log_r(z)\).

Would you like to explore any specific details or have any further questions?

Here are 5 related questions you might consider:
1. What is the value of \(z\) if \(r = 10\) and \(m = 2\)?
2. How can you solve for \(r\) if \(\log_r(z) = 4\) and \(z = 16\)?
3. What is the value of \(m\) if \(\log_5(z) = m\) and \(z = 25\)?
4. How can logarithms be used to solve exponential equations?
5. What is the difference between natural logarithms and common logarithms?

**Tip:** When working with logarithms, remember the change of base formula: \(\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\), which allows you to compute logarithms with any base using a different base, typically \(10\) or \(e\).

Explore how to solve logarithmic equations of the form log_r(z) = m. Learn the conversion to exponential form and apply it with examples. Suitable for Grades 10-12.

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