Problem 16 - Algebra 2 and Trigonometry

Question

Solve each equation and check. \(3^{x}=27\)

Step-by-Step Solution

Answer

The solution is \(x = 3\).

Step 1: Rewrite the Equation in Terms of the Same Base

Recognize that both sides of the equation can be expressed as powers of 3. Recall that 27 is a power of 3: \[27 = 3^3\]So, rewrite the equation \(3^x = 3^3\).

Step 2: Equate the Exponents

Since the bases are the same, you can equate the exponents. This gives the equation:\[x = 3\].

Step 3: Solve for x

The equation \(x = 3\) is already solved for \(x\). So, the solution is \(x = 3\).

Step 4: Verify the Solution

Substitute \(x = 3\) back into the original equation to check:\[3^3 = 27\]Since both sides are equal, the solution \(x = 3\) is verified.

Key Concepts

Base and Exponent

In exponential equations, we come across terms with a base and an exponent. The base is the number that is multiplied by itself a certain number of times. The number of times the base is multiplied is determined by the exponent. For example, in the term \(3^x\), the base is 3 and \(x\) is the exponent. This implies that 3 is multiplied by itself \(x\) number of times.
Understanding the role of base and exponent is crucial for solving exponential equations. When solving these types of problems, identifying a common base simplifies the process, allowing us to equate the exponents directly. In the initial exercise \(3^x = 27\), we rewrote 27 as \(3^3\), allowing us to focus directly on the exponents since both sides now had a common base of 3.

Powers of Numbers

A power of a number refers to how many times to use the number in a multiplication sequence. It's essentially what results from using a number as a base and an exponent to show repeated multiplication. For instance, \(3^3\), evaluated as 3 multiplied by itself three times, equals 27.
Knowing the powers of numbers can make recognizing patterns in exponential equations easier. Common powers, like \(2^3 = 8\) or \(5^2 = 25\), often serve as mental shortcuts, allowing students to 'see' the equivalence between terms without needing extensive calculations. Recognizing powers quickly is a helpful skill when simplifying or solving exponential equations.

Solving Equations

Solving equations, particularly exponential ones, involves finding the value of the variable that makes the equation true. Through our example \(3^x = 27\), we saw a practical method of solving: expressing both sides of the equation with the same base if possible.
Once we rewrote 27 as \(3^3\), the equation became \(3^x = 3^3\). Because the bases were already the same, the next step was simple. We could confidently equate the exponents since if \(a^b = a^c\), then we know \(b = c\).

• This approach works because we equate the powers when the bases match, simplifying the question to simply solving for the exponent, \(x\).
• Always verify solutions in the original equation, ensuring all calculations align and the solution satisfies the initial problem.

Understanding this process of rewriting in terms of a common base and equating exponents simplifies the task of solving exponential equations.