Problem 8 - Algebra for College Students

Question

Solve each of the equations. $$ 3^{x+1}=9 $$

Step-by-Step Solution

Answer

x = 1.

Step 1: Understanding the Equation

The equation given is \( 3^{x+1} = 9 \). The aim is to find the value of \( x \) that satisfies this equation. We notice that both sides of the equation can be expressed with the same base, which is 3.

Step 2: Rewriting the Right Side with Base 3

The number 9 can be rewritten as a power of 3. Since \( 9 = 3^2 \), we can express the equation as \( 3^{x+1} = 3^2 \).

Step 3: Equating the Exponents

Since the bases are the same (both are 3), we can set the exponents equal to each other: \( x + 1 = 2 \).

Step 4: Solving for x

To solve for \( x \), subtract 1 from both sides of the equation: \( x = 2 - 1 \). Thus, \( x = 1 \).

Key Concepts

Equation Solving

Solving equations is a fundamental skill in algebra. An equation is like a balance scale; what you do to one side, you must do to the other to keep both sides equal. In this exercise, we're given the exponential equation \( 3^{x+1} = 9 \). The goal here is to find the value of \( x \) that makes the equation true. When solving equations, always look for opportunities to simplify or restructure them for easier solving. This provides a solid start to tackle more complex mathematical problems.

Expressing with the Same Base

A powerful technique for solving exponential equations is expressing both sides with the same base. In our example, \( 3^{x+1} = 9 \), we notice that both 3 and 9 are powers of the number 3.

By rewriting 9 as \( 3^2 \), we harmonize the equation into a form \( 3^{x+1} = 3^2 \). This simplifies the process by focusing on comparing and equating the exponents instead of dealing with different bases.

• Identifying common bases helps to simplify equations significantly.
• This method is particularly useful when dealing with exponential equations, as it converts them into more manageable linear equations.

Remember, making both sides of the equation have the same base can be a game-changer in solving exponential problems efficiently.

Equating Exponents

Once both sides of an exponential equation are expressed with the same base, the next step is equating the exponents. This is because if the bases are the same, the exponents must be equal for the equation to hold true.

In the equation \( 3^{x+1} = 3^2 \), since both bases are 3, we set the exponents \( x + 1 \) and 2 equal to each other: \( x + 1 = 2 \).

• Equating exponents transforms the problem into a simple algebraic equation.
• This method relies on the property that if \( a^m = a^n \), then \( m = n \).

With this approach, the complexity of the equation is reduced, paving the way for the final step of solving for the variable.

Solving for Variable

The culmination of the previous steps leads us to solving for the variable \( x \). After equating the exponents, we end up with a simple equation like \( x + 1 = 2 \).

To isolate \( x \), we perform straightforward algebraic operations. Subtract 1 from both sides:

• \( x + 1 = 2 \)
• \( x = 2 - 1 \)

Thus, \( x = 1 \). Solving for the variable in this straightforward manner not only reinforces algebraic skills but also demonstrates how breaking down complex problems into smaller, manageable steps can simplify the path to the solution.