Solve the exponential equation 16 = 4 raised to (x-1) 

We are given the exponential equation:

\[
16 = 4^{x-1}
\]

### Step 1: Express both sides of the equation with the same base.
Notice that both 16 and 4 can be written as powers of 2:
\[
16 = 2^4 \quad \text{and} \quad 4 = 2^2
\]
Substituting \(4 = 2^2\) into the equation:

\[
16 = (2^2)^{x-1}
\]

### Step 2: Simplify the right-hand side.
By applying the power rule \((a^m)^n = a^{m \cdot n}\), we get:

\[
16 = 2^{2(x-1)}
\]
Simplifying the exponent on the right:

\[
16 = 2^{2x - 2}
\]

### Step 3: Set the exponents equal.
Since the bases are the same (both sides are powers of 2), we can set the exponents equal:

\[
4 = 2x - 2
\]

### Step 4: Solve for \(x\).
Now solve the equation for \(x\):

\[
4 + 2 = 2x
\]

\[
6 = 2x
\]

\[
x = 3
\]

Thus, the solution to the equation is:

\[
x = 3
\]

Would you like a detailed explanation of any specific step or further clarification?

Here are 5 related questions:
1. How do you solve an equation with different exponential bases?
2. Can you explain how to apply the power rule for exponents in more complex examples?
3. What is the general approach for solving exponential equations?
4. How do logarithms help in solving exponential equations?
5. What are some real-world applications of exponential equations?

**Tip**: When solving exponential equations, always try to rewrite both sides with the same base if possible—it simplifies the problem greatly!

Solve the exponential equation 16 = 4 raised to (x-1)

This page provides a step-by-step guide to solving the exponential equation 16 = 4^(x-1). The solution involves rewriting both sides with the same base and applying exponent rules. Suitable for students in Grades 9-11 learning about algebraic exponentiation and exponential equations.

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