Problem 15 - Algebra 2 and Trigonometry

Question

Solve each equation and check. \(2^{x}=16\)

Step-by-Step Solution

Answer

The solution is \(x = 4\).

Step 1: Write the equation with the same base

Recognize that 16 can be expressed as a power of 2. Rewrite the equation as follows:\[ 2^x = 2^4 \]Since both sides of the equation have the same base, you can now equate the exponents.

Step 2: Equate the exponents

Since the bases are equal, equate the exponents:\[ x = 4 \]This simplifies our equation to \( x = 4 \).

Step 3: Check the solution by substitution

Substitute \( x = 4 \) back into the original equation to verify the solution:\[ 2^x = 16 \Rightarrow 2^4 = 16 \]Calculate \( 2^4 \), which results in 16, confirming that the solution \( x = 4 \) is correct.

Key Concepts

Equating Exponents

When faced with an equation like \(2^x = 16\), one efficient way to solve it is by equating the exponents. First, you must express both sides of the equation with the same base. Here, we recognize that 16 can be rewritten as \(2^4\), since 16 is a power of 2. This transforms the equation to:

• \(2^x = 2^4\)

Now, both sides of the equation have the same base, which means we can equate the exponents. Essentially, this means that whatever exponent is on the left side must equal the exponent on the right side because the bases are identical. Thus, we can directly equate:

• \(x = 4\)

Equating exponents is a powerful method in simplifying equations when the bases are the same, allowing us to solve problems that might look complex at first glance.

Solving Exponential Equations

Solving exponential equations involves isolating the exponential expression and reducing it by equating powers, whenever possible. For an equation like \(2^x = 16\), the goal is to express both sides with the same base. A step-by-step process helps:

• Convert each side to the same base, if possible. Here, 16 becomes \(2^4\).
• Write down the equation with equivalent bases: \(2^x = 2^4\).
• Once bases match, focus on the exponents. Equate them: \(x = 4\).

After solving the equation, it is important to verify the result by substituting the solution back into the original equation to ensure correctness. In our example, substituting \(x = 4\) reinstates the original equation's validity:

• \(2^4 = 16\), which confirms \(x = 4\).

By routinely practicing these steps, solving exponential equations becomes simpler and more intuitive.

Power of a Number

Understanding the power of a number is key to grasping equations like \(2^x = 16\). The concept of power describes how many times to multiply the base by itself. In the expression \(2^4\), the base is 2, and the exponent is 4, indicating that 2 is multiplied by itself 4 times, i.e.,

• \(2 \times 2 \times 2 \times 2 = 16\)

This shows that 16 is a power of 2. When interpreting powers, remember:

• The base is the number we keep multiplying.
• The exponent tells us how many times we multiply the base.

Understanding this helps in identifying powers quickly in exponential equations. Whenever you're tackling exponential equations, remember to express numbers as powers of the same base wherever possible. This makes it easier to equate exponents directly and solve the equation.