Solution: Set $ 3^{t} = 9^{t-1} $. Since $ 9 = 3^{2} $, rewrite the equation as $ 3^{t} = (3^{2})^{t-1} $. Simplify the right-hand side: $ 3^{t} = 3^{2(t-1)} $. Equate the exponents: $ t = 2(t - 1) $. Solve: $ t = 2t - 2 $, so $ t = 2 $. Verify by substituting back: $ 3^{2} = 9^{1} $ → $ 9 = 9 $. Thus, the time is $ \boxed{2} $.
["Solve ( 3^t = 9^{t-1} ) – A Step-by-Step Algebraic Solution", "Equations involving exponents are common in math, science, and engineering, but solving them correctly requires careful manipulation. One classic example is solving the equation:", "[\n3^t = 9^{t-1}\n]", "At first glance, the bases differ—3 and 9—but this equation reveals a deeper relationship by expressing both sides with the same base. Let’s explore how to solve it step-by-step, with clear reasoning and verification.", "---", "### Step 1: Rewrite Both Sides with a Common Base", "Since ( 9 ) is a power of ( 3 ), rewrite ( 9 ) as ( 3^2 ):", "[\n3^t = (3^2)^{t-1}\n]", "This transformation is key: expressing both sides in terms of base 3 allows us to compare exponents directly.", "---", "### Step 2: Apply the Power of a Power Rule", "Use the exponent rule ( (a^m)^n = a^{m \cdot n} ) on the right-hand side:", "[\n3^t = 3^{2(t-1)}\n]", "Now both sides have the same base—3—so we can equate the exponents:", "[\nt = 2(t - 1)\n]", "---", "### Step 3: Solve the Simplified Equation", "Expand the right side:", "[\nt = 2t - 2\n]", "Subtract ( 2t ) from both sides:", "[\nt - 2t = -2 \quad \Rightarrow \quad -t = -2\n]", "Multiply both sides by -1:", "[\nt = 2\n]", "---", "### Step 4: Verify the Solution", "Substitute ( t = 2 ) back into the original equation to ensure correctness:", "Left-hand side:\n[\n3^2 = 9\n]", "Right-hand side:\n[\n9^{2 - 1} = 9^1 = 9\n]", "Both sides are equal, confirming the solution:", "[\n\boxed{2}\n]", "---", "### Why This Method Works", "This method leverages the unique property of exponential equations:\nIf ( a^m = a^n ) and ( a > 0 ), ( a <br/>\ne 1 ), then ( m = n ).\nBy converting everything to the same base, we simplify the problem from looking at entire growing functions to comparing simple numerical exponents—making the solution both elegant and reliable.", "---", "### Practical Applications", "Equations like ( 3^t = 9^{t-1} ) emerge in topics such as:\n- Exponential growth models in biology,\n- Compound interest calculations,\n- Radioactive decay problems,\n- And computer algorithms involving repeated exponentials.", "Understanding how to manipulate base exponents opens doors to solving more complex problems efficiently.", "---", "Final takeaway: With strategic substitution and exponent rules, exponential equations become manageable. The solution ( t = 2 ) not only solves the equation but illustrates a core principle in algebra—equality of exponents when bases match."]
