Let's analyze and solve the equation step-by-step:
Given the equation:
[tex]\[ 3^{m-1} = \frac{1}{3} \cdot 3^m \][/tex]
Step 1: Simplify the equation
First, we can rewrite [tex]\( \frac{1}{3} \cdot 3^m \)[/tex] using properties of exponents:
[tex]\[ \frac{1}{3} = 3^{-1} \][/tex]
Thus,
[tex]\[ \frac{1}{3} \cdot 3^m = 3^{-1} \cdot 3^m \][/tex]
Using the rule of exponents [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex], we get:
[tex]\[ 3^{-1} \cdot 3^m = 3^{-1+m} \][/tex]
So the equation becomes:
[tex]\[ 3^{m-1} = 3^{-1+m} \][/tex]
Step 2: Set the exponents equal
Since the bases on both sides of the equation are the same, we can set the exponents equal to each other:
[tex]\[ m - 1 = -1 + m \][/tex]
Step 3: Solve for [tex]\( m \)[/tex]
To solve for [tex]\( m \)[/tex], let's simplify the equation:
[tex]\[ m - 1 = -1 + m \][/tex]
Subtract [tex]\( m \)[/tex] from both sides to isolate the constants:
[tex]\[ m - 1 - m = -1 + m - m \][/tex]
Which simplifies to:
[tex]\[ -1 = -1 \][/tex]
This statement is always true and does not provide a specific value for [tex]\( m \)[/tex]. Therefore, it tells us that there is no unique solution for [tex]\( m \)[/tex]; rather, the equation does not limit [tex]\( m \)[/tex] to a particular value, indicating that no explicit solutions exist in this context.
Conclusion:
Given the steps above, the equation [tex]\( 3^{m-1} = \frac{1}{3} \cdot 3^m \)[/tex] does not yield any specific values for [tex]\( m \)[/tex]. Thus, the solution to this equation is the empty set:
[tex]\[ \boxed{[]} \][/tex]
