Problem 7 - Precalculus with Limits

Question

Solve for \(x.\) \(4^{x}=16\)

Step-by-Step Solution

Answer

The solution to the equation \(4^{x}=16\) is \(x=2\).

Step 1: Rewrite the Equation in the Same Base

Since the base on both sides of the equation must be the same to solve for \(x\), 16 can be converted into base 4 as \(4^2\). Therefore, the equation \(4^{x}=16\) can be rewritten as \(4^{x}=4^2\).

Step 2: Equate the Exponents

Now that both sides of the equation have the same base, the exponents must be equal as well. Hence, \(x=2\).

Key Concepts

Base Conversion

In the world of exponents, converting numbers to the same base can simplify solving equations. When you encounter an equation like \(4^x = 16\), it's helpful to express all terms using a common base. This is known as base conversion.

The goal is to transform terms to powers of the same number. For example, the number 16 can be expressed as \(4^2\). By rewriting 16 in this way, you create a common ground to compare the exponents. This makes the equation easier to solve.

If you can break down a number into smaller, equal parts, that's a clue it can be rewritten in a different base. Often, bases that are powers of 2, 3, or another common number work well. Practice converting between different bases will improve your skills in solving these equations.

Equating Exponents

Once an expression is rewritten in the same base, the puzzle becomes simpler. At this point, you focus solely on the exponents. This step is known as equating exponents.

In the equation \(4^x = 4^2\), the bases (the number 4) on both sides are the same. Therefore, the exponents must also be equal for the equality to hold. This principle stems from the basic property of exponents that if \(a^m = a^n\), then \(m = n\).

So, by directly comparing the exponents from each side of the equation, you can find the value of \(x\). Here, matching \(x\) with 2 is a straightforward task.

Solving Exponential Equations

Exponential equations, like \(4^x = 16\), feature variables as exponents. Solving these requires a keen eye for recognizing possible base conversions and applying the property of equating exponents.

These steps simplify solving such equations:

• Identify if both sides of the equation can be related by a common base.
• Convert numbers into that base to match both sides.
• Equate the exponents, and solve the resulting equation.

Start by guessing or observing possible bases when numbers are small or well known. Eventually, practice and familiarity will make the conversion and solving process quick and intuitive. This systematic approach makes exponential equations less intimidating and more approachable to solve.