Problem 21 - Algebra 2 and Trigonometry

Question

Solve each equation and check. \(6^{3 x}=6^{x-1}\)

Step-by-Step Solution

Answer

x = -1/2.

Step 1: Identify the Equation Structure

Examine the equation given as \(6^{3x} = 6^{x-1}\). Notice both sides have the same base, which is 6, making it possible to equate the exponents directly.

Step 2: Equate the Exponents

With both sides having the same base, we can write \(3x = x - 1\). The problem now reduces to solving this simple linear equation.

Step 3: Solve for x

Subtract \(x\) from both sides to isolate the term with \(x\) on one side: \(3x - x = -1\), which simplifies to \(2x = -1\).

Step 4: Divide to Find x

Now, divide both sides by 2 to solve for \(x\): \(x = \frac{-1}{2}\).

Step 5: Check the Solution

Substitute \(x = \frac{-1}{2}\) back into the original equation to verify: \(6^{3(-1/2)} = 6^{-1} = 6^{x-1}\). Calculate: \(6^{-3/2} = 6^{-1}\). Both sides are equal, confirming that \(x = \frac{-1}{2}\) is correct.

Key Concepts

Understanding Exponential Equations

Exponential equations are equations where the variable appears in the exponent. They form a fascinating part of algebra due to their unique properties. In our example equation, \(6^{3x} = 6^{x-1}\), both sides of the equation have a base of 6, which simplifies the process of solving the equation.

Such equations are often approached by employing the technique of base comparison, especially when the bases on both sides of the equation are the same. When faced with exponential equations, students should
- Look for common bases if possible. This can simplify the equation significantly.
- Remember that not all exponential equations will have the same base, but for those that do, solving them becomes much simpler by equating the exponents.
Exponential equations can model various real-world situations, such as population growth, radioactive decay, and compound interest. Understanding how to manipulate and solve them is fundamental for deeper mathematical studies.

Mastering Base Comparison

Base comparison is a useful method for solving exponential equations, particularly when both sides of the equation share the same base. In our example problem, \(6^{3x} = 6^{x-1}\), the base of both sides is 6.

This allows us to equate the exponents directly:

• Since the bases are identical, you can confidently set the exponents equal to each other: \(3x = x - 1\).
• This conversion turns the exponential equation into a much simpler linear equation - easier to solve.

This technique is especially helpful because it transforms a potentially complex equation into something more manageable. It's a strategy that, once understood, will significantly reduce the time and effort needed to solve similar problems.

Solving Linear Equations

Once an exponential equation is simplified into a linear equation, solving it becomes straightforward. A linear equation is one where the variable is not an exponent — it's your typical algebraic equation. Take the result from our base comparison step: \(3x = x - 1\).

The steps to solve this are:
- Subtract \(x\) from both sides to start simplifying: \(3x - x = -1\).
- This becomes \(2x = -1\).
- Divide each side by 2 to isolate \(x\): \(x = \frac{-1}{2}\).
Linear equations usually involve simple arithmetic operations such as addition, subtraction, multiplication, and division. They're often the stepping stone to checking your work with graphs or plugging back into the original equations to verify solutions. Always remember to cross-check your solution by plugging it back into the original equation — it confirms your approach and solution are correct.