Problem 276 - Calculus Volume 1

Question

For the following exercises, solve the exponential equation exactly. $$ 5^{x}=125 $$

Step-by-Step Solution

Answer

The solution is \(x = 3\).

Step 1: Express 125 as a power of 5

Rewrite 125 as a power of 5. Since \(125 = 5^3\), we express 125 in terms of base 5.

Step 2: Set the exponents equal

Since both sides of the equation \(5^x = 125\) can be expressed with the same base (base 5), we can set the exponents equal to each other. This gives us the equation \(x = 3\).

Step 3: Verify the Solution

To confirm, substitute \(x = 3\) back into the original equation: \(5^3 = 125\). Since this is true, our solution is verified.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. In our exercise, this concept plays a central role.

• The base is 5, and the task is to find what power of 5 will equal 125.
• The expression \(5^x\) signifies the multiplication of the base (5) repeatedly, for however many times the exponent (\(x\)) dictates.

In this case, we look for the value of \(x\) that satisfies this equation. Exponentiation allows us to simplify complex multiplication into a more compact form. It's like a shortcut for multiplying the same number over and over. This concept is vital in many fields such as algebra, calculus, and even computer science, where powers are used to represent data sizes, like in megabytes (\(10^6\)) or gigabytes (\(10^9\)). Understanding exponentiation helps simplify not only equations but various computational problems as well.

Equation Solving Techniques

Solving exponential equations often requires specific techniques that make the task more straightforward. In the given exercise, we used a technique based on expressing the number on the right side of the equation as a power of the base on the left side.The steps followed in our solution were:

• Rewrite the number: Transform the number 125 into \(5^3\). This step is crucial because having the same base on both sides simplifies the process.
• Same base method: Once both sides of the equation have the same base, you can equate the exponents directly. The equation \(5^x = 5^3\) allows us to equate \(x = 3\).

This technique simplifies solving exponential equations by reducing complex equations to simpler arithmetic tasks. Choosing the appropriate method when faced with different equations can make a big difference in solving problems efficiently. Being able to manipulate numbers to get the same base greatly aids in solving these equations straightforwardly. Remember, practice with various problems can strengthen understanding and application of these techniques.

Verification

Verification is crucial in mathematics, as it provides confirmation that the solution derived is indeed correct. For our exponential equation, we performed a verification step after finding the solution. Here’s how verification worked in our exercise:

• Substitute back: Once we found \(x = 3\), we substituted it back into the original equation \(5^x = 125\).
• Check equality: Calculating \(5^3\) to see if it equals 125, confirmed the solution since it's a true statement.

Verification acts as a safeguard against human error, allowing confirmation of the process involved and ensuring reliability of the results obtained.In more complex equations, this step is indispensable, as it reassures the solution's validity. For any equation solving, always remember to verify your result. It might seem like an extra step, but it's worth the certainty it provides.