Problem 22 - Intermediate Algebra

Question

Solve each equation. $$ 3^{x+1}=27 $$

Step-by-Step Solution

Answer

The solution is \( x = 2 \).

Step 1: Express 27 as a power of 3

The number 27 can be expressed as a power of 3. Since we know that \( 27 = 3^3 \), rewrite the equation using this information.

Step 2: Rewrite the equation using powers of 3

Now that we know \( 27 = 3^3 \), substitute this into the original equation to get: \[ 3^{x+1} = 3^3 \]

Step 3: Equate the exponents

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

Step 4: Solve for x

Subtract 1 from both sides of the equation to solve for \( x \): \[ x = 3 - 1 \] \[ x = 2 \]

Key Concepts

Solving Exponential Equations

When you come across an equation like \(3^{x+1} = 27\), you are dealing with an exponential equation. These types of equations can be tricky because the variable, in this case \(x\), is in the exponent. To solve them, the main goal is to manipulate the equation so that both sides have the same base. Once the bases match, you can directly equate the exponents and solve for the unknown variable.

Here's a simplified way to approach it:

• First, express all numbers as powers of the same base if possible. In our example, both sides of the equation should be expressed as a power of 3.
• Rewrite the equation with the same base on both sides. This turns our equation into \(3^{x+1} = 3^3\).
• Once both sides have the same base, simply equate the exponents: \(x+1 = 3\).
• Solve the resulting simple equation for the variable \(x\).

Using these steps will help you systematically approach and solve exponential equations.

Using Properties of Exponents

Exponents have a few crucial properties that make solving exponential equations more manageable. Knowing these properties allows you to manipulate the equations when looking for solutions.

Some key properties include:

• Multiplying Powers with the Same Base: \(a^m \times a^n = a^{m+n}\)
• Dividing Powers with the Same Base: \(a^m \div a^n = a^{m-n}\)
• Power of a Power: \((a^m)^n = a^{m \times n}\)

Understanding these properties helps in rewriting the exponential equation into a form where exponent conflicts are easily resolved.

In our example, we expressed 27 as \(3^3\) using the power of a number property. This allowed us to shift the equation into a form where the bases match, simplifying further steps.

Algebraic Manipulation

Algebraic manipulation is a strategy used to simplify equations systematically. It involves using algebraic methods to transform the equation into a basic form that is easier to solve.

Here's how you apply it to solve the exponential equation \(3^{x+1} = 3^3\):

• Since both sides have the same base, you equate the exponents directly: \(x+1 = 3\).
• Next, solve for the variable \(x\) by isolating it. Subtract 1 from both sides to get: \(x = 3 - 1\).
• Complete the arithmetic to find that \(x = 2\).

This step-by-step breakdown demystifies the process, making it easier to solve not just this equation, but applying similar methods for other exponential equations too.