What is the general method for solving ax=b*ax*=*b*?

To solve the general exponential equation \(a^x = b\), where \(a > 0\) and \(a \neq 1\), follow these steps:

### 1. **Take the logarithm of both sides:**
   Apply the logarithm to both sides of the equation. You can use any logarithm, but it's common to use the natural logarithm (\(\ln\)) or the common logarithm (\(\log\)). For this explanation, we'll use the natural logarithm.

   \[
   \ln(a^x) = \ln(b)
   \]

### 2. **Use the power rule of logarithms:**
   The power rule states that \(\ln(a^x) = x \cdot \ln(a)\). Apply this to the left side:

   \[
   x \cdot \ln(a) = \ln(b)
   \]

### 3. **Solve for \(x\):**
   Now, isolate \(x\) by dividing both sides by \(\ln(a)\):

   \[
   x = \frac{\ln(b)}{\ln(a)}
   \]

### Example:
Let's say you have \(3^x = 81\). To solve it:

1. Take the logarithm of both sides:

   \[
   \ln(3^x) = \ln(81)
   \]

2. Use the power rule:

   \[
   x \cdot \ln(3) = \ln(81)
   \]

3. Solve for \(x\):

   \[
   x = \frac{\ln(81)}{\ln(3)}
   \]

   Since \(81 = 3^4\), \(x = 4\).

Would you like more details or have other questions?

Here are 8 related questions you might find useful:

1. How can you solve \(2^x = 50\) using logarithms?
2. What is the difference between using natural logarithms and common logarithms in these equations?
3. How would you solve \(5^x = 100\)?
4. Can you solve \(a^x = b\) if \(a\) or \(b\) is a negative number?
5. What happens if \(a = 1\) in the equation \(a^x = b\)?
6. How do you solve exponential equations with different bases, like \(2^x = 3^y\)?
7. How would you solve an equation like \(3^{2x} = 81\)?
8. What is the method for solving \(e^x = 7\), where \(e\) is the base of natural logarithms?

**Tip:** When solving exponential equations, always consider the properties of logarithms and how they can simplify the process.

Learn how to solve the exponential equation a^x = b using logarithms with a detailed step-by-step solution. This problem covers key concepts in exponential equations and logarithms. Suitable for Grades 10-12 students and anyone interested in mathematical problem-solving.

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