give the step by step to solve exponential inequalities and explain 

To solve exponential inequalities, follow these steps:

### Step 1: **Isolate the Exponential Expression**
   - Begin by isolating the exponential term on one side of the inequality. For example, if you have \(2^x > 8\), ensure that \(2^x\) is alone on one side.

### Step 2: **Rewrite the Inequality with a Common Base (if possible)**
   - If possible, rewrite the inequality so that both sides have the same base. For instance, \(2^x > 8\) can be rewritten as \(2^x > 2^3\).
   - If rewriting with a common base is not feasible, consider using logarithms.

### Step 3: **Apply Logarithms (if needed)**
   - If you can't rewrite the inequality with a common base, apply logarithms to both sides. Using the natural logarithm (\(\ln\)) or common logarithm (\(\log\)) works. For example, if you have \(3^x < 20\), take the logarithm of both sides: \(\log(3^x) < \log(20)\).

### Step 4: **Solve the Inequality**
   - If you have the same base on both sides, set the exponents equal to each other. For example, \(2^x > 2^3\) leads to \(x > 3\).
   - If you've taken logarithms, use the properties of logarithms to solve for the variable. For example, \(\log(3^x) = x \log(3)\), so \(x \log(3) < \log(20)\) leads to \(x < \frac{\log(20)}{\log(3)}\).

### Step 5: **Check the Direction of the Inequality**
   - When multiplying or dividing by a negative number, the inequality direction changes. However, when dealing with exponential functions, this issue usually arises if you take the logarithm of a negative base, which is undefined. Ensure the base of the exponential function is positive and not equal to 1.
   - If the exponential function is decreasing (base between 0 and 1), remember that the inequality direction reverses.

### Step 6: **Write the Solution in Interval Notation**
   - Express your solution in interval notation. For \(x > 3\), the solution in interval notation is \( (3, \infty) \).

### Example
Solve \(2^x > 8\):
1. Isolate \(2^x\): It's already isolated.
2. Rewrite with a common base: \(2^x > 2^3\).
3. Set the exponents: \(x > 3\).
4. Solution in interval notation: \( (3, \infty) \).

### Summary:
- Isolate the exponential expression.
- Rewrite with a common base or use logarithms.
- Solve the inequality.
- Be mindful of the inequality direction.
- Write the solution in interval notation.

Would you like more details on any of these steps, or do you have further questions?

### Related Questions:
1. How do you solve exponential inequalities involving natural exponent \(e^x\)?
2. What happens when the exponential inequality has multiple exponential terms?
3. How do you handle inequalities like \(e^{x-2} \leq 5\)?
4. What are some common mistakes to avoid when solving exponential inequalities?
5. How can you verify the solution of an exponential inequality?

### Tip:
When using logarithms to solve inequalities, remember that \(\log(x)\) is only defined for \(x > 0\). Make sure the arguments of all logarithms are positive.

give the step by step to solve exponential inequalities and explain

Learn how to solve exponential inequalities with a detailed step-by-step guide. This explanation covers key algebraic concepts such as isolating the exponential expression, using logarithms, and solving inequalities, along with interval notation. Suitable for students in Grades 9-12.

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