Problem 23 - Precalculus with Limits

Question

Using the One-to-One Property In Exercises \(23-26\) use the One-to-One Property to solve the equation for \(x .\) $$3^{x+1}=27$$

Step-by-Step Solution

Answer

The solution to the equation \(3^{x+1}=27\) is \(x = 2\).

Step 1: Express 27 as a power of 3

First, rewrite 27 as \(3^3\). Since 27 is equivalent to \(3^3\), we can substitute 27 with \(3^3\), so the equation becomes \(3^{x+1} = 3^{3}\).

Step 2: Apply the One-to-One Property

According to the One-to-One Property, if \(a^m = a^n\), then \(m = n\). Thus, we can equate the exponents, giving \(x + 1 = 3\).

Step 3: Solve for x

Solve the equation \(x + 1 = 3\) for x by subtracting 1 from both sides to get \(x = 3 - 1\).

Key Concepts

Exponential Equations

Exponential equations are equations where the variable appears in an exponent. This is different from linear or quadratic equations, where the variable might appear in a more straightforward manner. Understanding how to work with exponential equations requires learning specific techniques, like the One-to-One Property, to solve these types of equations effectively.
To solve an exponential equation, the first step is usually to express each side with the same base. For example, in the equation \(3^{x+1} = 27\), we recognize that 27 can be rewritten as \(3^3\). By expressing both sides of the equation with the base of 3, the problem becomes much easier to handle.
This transformation allows us to take advantage of properties of exponents, which simplifies the equation into a format that's more straightforward to solve. Once you can compare the exponents equally, you're well on your way to finding the solution.

Properties of Exponents

The properties of exponents are fundamental rules that help simplify and solve exponential expressions. These rules are incredibly helpful when solving exponential equations or simplifying expressions.

• **Product of Powers Property**: If you have the same base multiplying, you can add the exponents. For example, \(a^m \cdot a^n = a^{m+n}\).
• **Quotient of Powers Property**: When dividing like bases, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• **Power of a Power Property**: When raising an exponent to another exponent, multiply them: \((a^m)^n = a^{m \cdot n}\).
• **Zero Exponent Rule**: Any base raised to the power of zero equals one: \(a^0 = 1\) (as long as \(a eq 0\)).

In the context of our given equation \(3^{x+1} = 3^{3}\), we simply use the One-to-One Property. It is derived from understanding that if the bases are the same, their exponents must also be equal, thus \(x+1=3\). Understanding these rules makes solving equations much more intuitive.

Solving Equations

Solving equations, especially those with exponents, involves a series of logical steps and a good grasp of mathematical principles.
Starting with the equation \(3^{x+1} = 3^{3}\), recognizing it as a solvable form is key. This involves understanding how to manipulate the equation using the properties of exponents and focusing on isolating the variable to find its value.
Once you apply the One-to-One Property to equate \(x + 1\) to \(3\), what remains is a basic linear equation. Solving \(x + 1 = 3\) involves simple arithmetic: subtract 1 from both sides to isolate \(x\), leading to \(x = 2\).

• Always begin by expressing all terms with like bases, if possible.
• Keep the equation balanced by performing the same operations on each side.
• Simplify the equation step-by-step to isolate the variable.

Mastering these steps will enable you to confidently tackle a wide range of exponential equations.