We need to solve the equation \(5^{(x - 9)} = 25^{(3 - x)}\).

### Step 1: Express both sides with the same base
Notice that \(25\) can be written as \(5^2\). So, we rewrite the equation:

\[
5^{(x - 9)} = (5^2)^{(3 - x)}
\]

### Step 2: Simplify the exponents
Using the exponentiation rule \((a^m)^n = a^{m \cdot n}\), we simplify the right-hand side:

\[
5^{(x - 9)} = 5^{2 \cdot (3 - x)}
\]

\[
5^{(x - 9)} = 5^{(6 - 2x)}
\]

### Step 3: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal:

\[
x - 9 = 6 - 2x
\]

### Step 4: Solve for \(x\)
Add \(2x\) to both sides to collect the \(x\) terms on one side:

\[
x + 2x - 9 = 6
\]

\[
3x - 9 = 6
\]

Add \(9\) to both sides:

\[
3x = 15
\]

Divide by \(3\):

\[
x = 5
\]

So, the solution is \( \boxed{5} \).

Would you like further details on any step?

---

Here are 5 related questions:
1. How do you solve an equation with exponents if the bases are not the same?
2. What are the rules of exponentiation?
3. How can you express powers in terms of logarithms?
4. What is the relationship between exponents and logarithms?
5. How would the solution change if the base was different, such as \(7\) instead of \(5\)?

**Tip:** When solving exponential equations, always try to express both sides with the same base if possible. This simplifies the equation and makes solving for the variable straightforward.

Learn how to solve the exponential equation 5^(x - 9) = 25^(3 - x) step-by-step with detailed explanations. This problem involves exponentiation rules and logarithmic expressions. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation