Problem 103 - Algebra and Trigonometry

Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Solve: \(3^{2 x-3}=9^{1-x}\)

Step-by-Step Solution

Answer

x = \frac{5}{4}

Step 1: Rewrite the Equation with the Same Base

Start by expressing both sides of the equation with the same base. Note that 9 can be written as a power of 3, specifically, 9 = 3^2. So, rewrite the equation: \[3^{2x-3} = (3^2)^{1-x}\]

Step 2: Simplify the Right Side Using Exponent Rules

Apply the exponent rule \((a^m)^n = a^{mn}\) to simplify the right-hand side: \[3^{2x-3} = 3^{2(1-x)}\]

Step 3: Set the Exponents Equal to Each Other

Since the bases are the same, set the exponents equal to each other: \[2x - 3 = 2(1 - x)\]

Step 4: Solve the Exponential Equation

Expand the right-hand side and solve for x: \[2x - 3 = 2 - 2x\]Add 2x to both sides: \[2x + 2x - 3 = 2\]Simplify: \[4x - 3 = 2\]Add 3 to both sides: \[4x = 5\]Divide by 4: \[x = \frac{5}{4}\]

Key Concepts

Rewrite Equations with Same Base

When solving exponential equations, it's often useful to rewrite each term with the same base. This makes it easier to compare and solve.
For instance, in the problem given, we start with:
\(3^{2x-3}=9^{1-x}\).
Notice that 9 is a power of 3. Remember that 9 is equal to 3 squared, i.e., \(9 = 3^2\).
Rewriting the equation using this same base helps immensely.
The rewritten equation looks like this:
\(3^{2x-3} = (3^2)^{1-x}\).
Now, both sides of the equation have the same base.

Exponent Rules

Exponent rules are essential tools when working with exponential equations. They offer a way to simplify expressions and solve equations more easily.
Some key exponent rules include:

• Product Rule: \(a^m \cdot a^n = a^{m+n}\)
• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a Power Rule: \((a^m)^n = a^{mn}\)
• Zero Exponent Rule: \(a^0 = 1\) where a ≠ 0

In our problem, we used the Power of a Power Rule. Specifically, we simplified: \( (3^2)^{1-x} = 3^{2(1-x)}\).
This rule helps convert the nested exponents into a single exponent for easier comparison. Now our equation looks like: \(3^{2x-3} = 3^{2(1-x)}\).

Solving Exponential Equations

After rewriting the equation with the same base and applying exponent rules, we need to solve for the variable.
Here’s the simplified equation: \3^{2x-3} = 3^{2(1-x)}\
Since both sides have the same base, we can set their exponents equal to each other:
\2x - 3 = 2(1 - x)\
Now, solve for x:
Expand the right-hand side: \2x - 3 = 2 - 2x\.
Add \2x\ to both sides: \2x + 2x - 3 = 2\.
Simplify: \4x - 3 = 2\.
Add 3 to both sides: \4x = 5\.
Finally, divide by 4: \x = \frac{5}{4}\.
So, the solution to the exponential equation is \(x = \frac{5}{4}\).
With practice, these steps become second nature, and solving similar problems gets easier.