Problem 8 - Precalculus

Question

Solve the exponential equations exactly for \(x\). $$125^{x}=5^{2 x-3}$$

Step-by-Step Solution

Answer

The solution is \(x = -3\).

Step 1: Rewrite 125 and 5 as Powers of the Same Base

Recognize that both 125 and 5 are powers of 5. We can write 125 as \(5^3\). Therefore, the equation \(125^x = 5^{2x-3}\) can be rewritten as \((5^3)^x = 5^{2x-3}\).

Step 2: Apply the Power of a Power Rule

The power of a power rule states that \((a^m)^n = a^{m \cdot n}\). Apply this to \((5^3)^x\) to get \(5^{3x}\). The equation becomes \(5^{3x} = 5^{2x-3}\).

Step 3: Set the Exponents Equal

Since the bases are now the same \((5)\), set the exponents equal to each other: \(3x = 2x - 3\).

Step 4: Solve for x

Subtract \(2x\) from both sides of the equation: \[3x - 2x = -3\]This simplifies to \(x = -3\).

Key Concepts

Power of a Power Rule

The power of a power rule is a fundamental property of exponents. In simple terms, it states that when you raise an exponent to another exponent, you separate the exponents by multiplication. The rule can be written as \[ (a^m)^n = a^{m \cdot n} \]where:

• \(a\) represents the base.
• \(m\) and \(n\) are the exponents.

To apply this rule, focus on nesting exponents. Always multiply the exponents when you see parentheses enclosing a power raised to another power. In the given problem, \((5^3)^x\) was transformed into a single power: \(5^{3x}\). This multiplication step simplifies calculations and makes exponential equations easier to solve.

Solving for x

Solving for \(x\) involves isolating the variable on one side of the equation. This is achieved by using basic algebraic principles such as addition, subtraction, multiplication, or division. After applying the power of a power rule, the given equation simplifies to \[5^{3x} = 5^{2x-3}\]Since both sides share the same base (\(5\)), the exponents themselves must be equal. Hence, we equate the exponents:\[3x = 2x - 3\]To isolate \(x\), subtract \(2x\) from both sides:

• \(3x - 2x = x\)
• This gives us \(x = -3\)

Each step moves us closer to a clear value, simplifying the equation step by step.

Rewriting Bases

Rewriting bases in exponential equations helps in matching the powers conveniently. This process involves expressing numbers as powers of a common base. In the exercise, the numbers \(125\) and \(5\) needed to be compared. Recognizing the common base is crucial: both numbers can be written as powers of \(5\).

• \(125 = 5^3\)

Converting numbers to the same base facilitates easier equation manipulation since the exponents can be directly compared once the bases match. This strategic conversion is vital in solving exponential equations effectively. Once bases are harmonized, other properties of exponents, such as the power of a power rule, can be applied seamlessly, aiding in the resolution of the equation.