Problem 12 - College Algebra Essentials

Question

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$125^{x}=625$$

Step-by-Step Solution

Answer

After simplifying the given equation, rearranging and solving for x, the solution to the equation is \(x = 4/3\).

Step 1: Rewrite the equation

The given equation is \(125^x = 625\). Both 125 and 625 can each be written as a power of 5. Where \(125 = 5^3\) and \(625 = 5^4\). So substitute \(5^3\) for 125 and \(5^4\) for 625 to rewrite the equation as \((5^3)^x = 5^4\)

Step 2: Simplify the left side

In the case \((5^3)^x\), the rule of exponents states that when you raise a power to a power, you should multiply the exponents. Thus, you can simplify the left-hand side of the equation, resulting in \(5^{3x} = 5^4\)

Step 3: Equate the exponents

Both sides of the equation are now powers of 5. This means that for the equation to be true, the exponents also have to be equal. Therefore, set 3x equal to 4 to get the equation \(3x = 4\).

Step 4: Solving for x

To solve for x, divide each side by 3 to isolate the variable. This returns: \(x = 4/3\).

Key Concepts

Power of the Same Base

When solving exponential equations, expressing each term as a power of the same base can simplify the problem considerably. Imagine you're given an equation like \(125^x = 625\). Both numbers, 125 and 625, can be rewritten using a common base.
The trick is to express them in terms of powers of 5. For instance:

• 125 is \(5^3\)
• 625 is \(5^4\)

This conversion is crucial because it allows us to transform the equation into \((5^3)^x = 5^4\). With both sides expressed as powers of the same base, solving becomes much clearer and straightforward. Aligning with a single base lets you focus on equating exponents, which is a key technique for solving such equations.

Equating Exponents

Once both sides of an exponential equation are rewritten as powers with the same base, you can directly equate the exponents. This step leverages the property that if two exponential expressions with the same base are equal, their exponents must also be equal.
Now, in the equation \((5^3)^x = 5^4\), simplify by applying the rule of exponents to get \(5^{3x} = 5^4\). Here, since the base (5) is the same on both sides, you can confidently set the exponents equal:

• \(3x = 4\)

This is a pivotal step as it transforms the exponential equation into a simple linear equation, making it easier to solve for the variable \(x\). By equating the exponents, you make full use of the identical bases to find the solution.

Rule of Exponents

The rule of exponents is a powerful tool when dealing with exponential equations. One important rule involves powers raised to another power. According to this rule, you multiply the exponents. For example, in a situation like \((5^3)^x\), you use the rule of exponents to simplify: multiply 3 and \(x\) to get \(5^{3x}\).
This simplification is crucial as it turns complex expressions into manageable forms. Another rule often used is when identical bases are equal, indicating their exponents must be equal, which leads directly to a simple algebraic equation.
Solving exponential problems requires a solid understanding of these rules, as they provide shortcuts to solve for unknown variables efficiently. By using these rules, you not only simplify the operations but also make the equation intuitive and straightforward to solve.