Problem 13 - Precalculus: A Concise Course

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: Express Numbers as Power of 2

First, rewrite the numbers 4 and 16 as powers of 2. The number 4 can be expressed as \(2^2\), while 16 can be expressed as \(2^4\). So the equation \(4^{x} = 16\) can be expressed as \((2^2)^x = 2^4\).

Step 2: Simplify Left Side of the Equation

Next, simplify the left side of the equation. Because of the property of powers that states that \((a^m)^n = a^{mn}\), the left side of the equation can be rewritten as \(2^{2x}\), resulting in the equation \(2^{2x} = 2^4\).

Step 3: Compare the Exponents

Since the bases on both sides of the equation are the same (2), you can equate the exponents to one another, resulting in the equation \(2x = 4\).

Step 4: Solve for \(x\)

Finally, solve the equation \(2x = 4\) for \(x\) by dividing both sides of the equation by 2. This gives \(x = 4/2 = 2\).