Problem 24 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the exponential equation. $$3^{x}=243$$

Step-by-Step Solution

Answer

The solution to the equation \(3^x = 243\) is \(x = 5\).

Step 1: Express the Number in the Base Form

Express 243 in terms of the base 3. This is done because 243 can be expressed as \(3^5\). So, the equation now is \(3^x = 3^5\).

Step 2: Compare the Exponents

Once we expressed both sides of the equation in terms of the same base (3), we can simply compare the exponents. So, we set \(x = 5\).

Step 3: Confirm the Solution

Substitute x back into the original equation to confirm. If the left-side equals to the right-side, then it proves that the solution is correct. Plugging \(x = 5\) into the original equation gives \(3^5 = 243\), which is a true statement.

Key Concepts

Solving Exponential Equations

Exponential equations are equations where the variable appears in the exponent. Solving these equations usually requires you to isolate the term with the exponent. The objective is to express both sides of the equation with the same base. When the bases are equal, it becomes easy to solve for the exponent by comparing.Here are some key steps you can follow:

• Identify the base and exponent in the equation. For example, in the equation \(3^x = 243\), the base is 3 and the exponent is \(x\).
• Try to express the known number (in this case, 243) in terms of the base. We find that 243 can be expressed as \(3^5\).
• Once you have matched the bases, equate the exponents and solve for the variable.

By ensuring that both sides have the same base, you simplify the problem significantly, allowing for an easier solution.

Comparing Exponents

After expressing the terms in the exponential equation with the same base, the next step is to compare the exponents. The main principle here is that if the bases are the same on both sides of the equation, the exponents must also be equal. In the exercise, once 243 was expressed as \(3^5\), the equation \(3^x = 3^5\) can be solved by equating the exponents to get \(x = 5\).

• By comparing exponents, we use the fact that if \(a^m = a^n\), then \(m = n\).
• This step reduces the problem into a simple algebraic equation that is often straightforward to solve.

In this way, comparing exponents becomes the crucial step in finding the value of the variable.

Base Conversion in Exponentials

Converting numbers to the same base is a key strategy when solving exponential equations. This involves expressing a number as a power of another number. Base conversion helps to simplify the problem by focusing only on the exponents. Here's how to approach base conversion:

• Determine if the numbers can be expressed with the same base. For example, below are some characteristics:
  • Powers of 2: 4, 8, 16, 32, etc.
  • Powers of 3: 9, 27, 81, 243, etc.
  • Powers of 5: 25, 125, etc.
• Once you identify a common base, express each number using this base.
• This conversion allows you to easily equate the exponents once the bases are equal.

Using base conversion simplifies the equation and clarifies the relationship between numbers, thus making it much easier to solve the equation.