Question

If \( 3^x = 9^{x-1} \), then the value of \( x \) is:

• A. 2
• B. 1
• C. 0
• D. 3

Hint:

When solving exponential equations, convert the bases to be the same, then equate the exponents to find the solution.

Answer 1

The Correct Option is A

We are given the equation: \[ 3^x = 9^{x-1} \] Now, \( 9 \) can be rewritten as \( 3^2 \), so the equation becomes: \[ 3^x = (3^2)^{x-1} \] Using the rule \( (a^m)^n = a^{mn} \), we can simplify the equation: \[ 3^x = 3^{2(x-1)} \] Since the bases are the same, we can set the exponents equal to each other: \[ x = 2(x - 1) \] Now, solve for \( x \): \[ x = 2x - 2 \] \[ x - 2x = -2 \] \[ -x = -2 \] \[ x = 2 \] Thus, the value of \( x \) is 2.