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Chapter 11: Problem 2

For Problems \(1-34\), solve each equation. $$ 2^{2 x}=16 $$

Short Answer

Expert verified
The solution to the equation is \(x = 2\).

Step by step solution

01

Understand the Equation

The equation given is \(2^{2x} = 16\). We need to solve for \(x\). Notice that 16 can be rewritten as a power of 2.
02

Rewrite 16 as a Power of 2

Since 16 is equivalent to \(2^4\), we rewrite the equation as \(2^{2x} = 2^4\).
03

Equate the Exponents

Since the bases are the same, we can equate the exponents: \(2x = 4\).
04

Solve for x

To solve for \(x\), divide both sides of the equation by 2: \(x = \frac{4}{2} = 2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Equations
Solving equations is a fundamental skill in algebra. It involves finding the value of an unknown variable that makes the equation true. In our example, we started with the equation \(2^{2x} = 16\). The goal is to find the value of \(x\) that satisfies this equation.
Let's break down the process:
  • Identify the equation and understand what is required to solve it. Here, we need to find \(x\).
  • Notice patterns or similarities to make solving easier. For this equation, recognizing that 16 is a power of 2 significantly simplifies the problem.
  • Solve step-by-step while maintaining the equality. You adjust each side of the equation equally until you isolate the variable.
By following these steps, we ensure that we're systematically finding the correct solution.
Exponentiation
Exponentiation is a mathematical operation involving numbers known as "bases" and "exponents." Essentially, exponentiation is repeated multiplication of a number by itself. For example, in \(2^{2x}\), 2 is the base, and \(2x\) is the exponent.
Key points to understand about exponentiation include:
  • It represents repeated multiplication. For example, \(2^4 = 2 \times 2 \times 2 \times 2 = 16\).
  • Exponentiation follows certain rules, such as \(a^m \times a^n = a^{m+n}\) and \((a^m)^n = a^{mn}\).
  • In solving equations, rewriting terms as powers of a common base can simplify the problem. This is what we did by expressing 16 as \(2^4\).
Understanding how exponentiation works is critical in solving exponential equations effectively.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations or expressions using algebraic rules. This allows us to isolate variables and solve equations. In our solved equation, the crucial step was equating the exponents.
Here's how algebraic manipulation is applied:
  • Reformulate the equation when possible. The equation \(2^{2x} = 2^4\) allowed us to equate the exponents directly since the bases were the same.
  • Use inverse operations to isolate the variable. In our example, dividing both sides of \(2x = 4\) by 2 isolated \(x\).
  • Apply consistent operations to both sides of the equation to maintain equality. This ensures that the equation remains balanced.
Mastering algebraic manipulation is key to solving not just exponential equations, but a wide array of algebraic problems.

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Most popular questions from this chapter

For Problems \(11-20\), use your calculator to find \(x\) when given \(\log x\). Express answers to five significant digits. $$ \log x=-2.1928 $$

For Problems \(15-22\), solve each logarithmic equation. $$ \ln (3 t-4)-\ln (t+1)=\ln 2 $$

For a certain strain of bacteria, the number present after \(t\) hours is given by the equation \(Q=Q_{0} e^{0.34 t}\), where \(Q_{0}\) represents the initial number of bacteria. How long will it take 400 bacteria to increase to 4000 bacteria?

For Problems \(1-14\), solve each exponential equation and express solutions to the nearest hundredth. $$ 3 e^{x}=35.1 $$

Graph \(f(x)=\log _{4} x\) by reflecting the graph of \(g(x)=4^{x}\) across the line \(y=x\).
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