Set the following exponential equations equal to each other: 3^(x-9) = 81^(5x+1)

To solve the equation \( 3^{x-9} = 81^{5x+1} \), we first express both sides using the same base.

### Step 1: Rewrite \( 81 \) as a power of 3.
Since \( 81 = 3^4 \), we can rewrite the right-hand side of the equation:

\[
81^{5x+1} = (3^4)^{5x+1}
\]

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

\[
(3^4)^{5x+1} = 3^{4(5x+1)} = 3^{20x+4}
\]

### Step 3: Set the exponents equal to each other.
Now that both sides have the same base, we can set the exponents equal to each other:

\[
x - 9 = 20x + 4
\]

### Step 4: Solve for \( x \).
Isolate \( x \) by first subtracting \( x \) from both sides:

\[
-9 = 19x + 4
\]

Next, subtract 4 from both sides:

\[
-13 = 19x
\]

Finally, divide both sides by 19:

\[
x = \frac{-13}{19}
\]

Thus, the solution is:

\[
x = \frac{-13}{19}
\]

Would you like further details on the steps, or do you have any other questions?

Here are some related questions to explore:

1. How do you solve exponential equations with different bases?
2. What are the properties of exponents used in solving exponential equations?
3. How can logarithms be used to solve exponential equations?
4. What is the difference between exponential growth and exponential decay in equations?
5. Can you graph \( y = 3^{x-9} \) and \( y = 81^{5x+1} \) to visualize their intersection point?

**Tip:** Always try to express both sides of an exponential equation with the same base to simplify the solution process.

Discover how to solve the exponential equation 3^(x-9) = 81^(5x+1) by expressing both sides using the same base. This solution involves properties of exponents and step-by-step simplification, making it suitable for Grades 9-11 students.

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