Problem 6 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. Do not use a calculator. $$125^{x}=5$$

Step-by-Step Solution

Answer

The solution is \( x = \frac{1}{3} \).

Step 1: Express 125 and 5 as Powers of 5

Identify how the numbers 125 and 5 can be expressed as powers of the same base, which in this case is 5. Observe that \(125 = 5^3\) and \(5 = 5^1\). This allows us to rewrite the equation with a common base.

Step 2: Rewrite the Equation

Using the expressions for 125 and 5 obtained in the first step, rewrite the given equation in terms of powers of 5:\[ (5^3)^x = 5^1 \]

Step 3: Apply the Power of a Power Rule

Apply the rule of exponents that states \((a^m)^n = a^{mn}\). With this, you can simplify the left side of the equation:\[ 5^{3x} = 5^1 \]

Step 4: Equate the Exponents

Since the bases are the same, equate the exponents from both sides of the equation:\[ 3x = 1 \]

Step 5: Solve for x

Solve the equation \(3x = 1\) by dividing both sides by 3:\[ x = \frac{1}{3} \]

Key Concepts

Exponent Rules

Exponent rules are fundamental to working with exponential equations. They include several properties that allow us to simplify expressions involving powers. Here are some key points to remember:

• Product of powers: To multiply two expressions with the same base, add their exponents: \( a^m \cdot a^n = a^{m+n} \).


• Quotient of powers: To divide two expressions with the same base, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).


• Power of a power: To raise a power to another power, multiply the exponents: \( (a^m)^n = a^{m \cdot n} \).


• Zero exponent rule: Any nonzero number raised to the power of zero equals one: \( a^0 = 1 \).

In the given exercise, these rules allow us to simplify complex expressions into a manageable form to solve the exponential equation.

Common Base

The common base concept is essential for solving exponential equations effectively. When different numbers are written as powers but need to be compared or simplified, using a common base simplifies this process.

• Start by identifying a suitable base that both numbers can be expressed as powers of.


• For our problem, 125 and 5 are both powers of 5, i.e., \(125 = 5^3\) and \(5 = 5^1\).

Writing these numbers with a common base helps in aligning the exponents, thereby making it possible to equate them easily. This step is essential as it transforms the original equation into a form where exponent rules can be efficiently applied.

Power of a Power Rule

The power of a power rule is another critical tool in simplifying equations involving exponents. This rule states that when you raise a power to another power, you multiply the exponents.
In the context of our exercise, this rule allows for the transformation of the equation \((5^3)^x = 5^1\) into a simpler form: \(5^{3x} = 5^1\). By doing so, it reduces a potentially cumbersome exponent expression into a more straightforward one by focusing on the exponentiation process.
Once simplified using this rule, the path to solving for the unknown variable "x" becomes clearer: equate the exponents, which results in the simple equation \(3x = 1\). From here, solving for \(x\) becomes straightforward, leading to the final answer \(x = \frac{1}{3}\). Thus, understanding and applying the power of a power rule is pivotal in managing and solving exponential equations.