Problem 32 - Algebra 2

Question

Solve each equation. If necessary, round to the nearest thousandth. $$ 2^{x}=4 $$

Step-by-Step Solution

Answer

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

Step 1: Identify common base

Note that \(2^{x}=4\) can be written as \(2^{x}=2^{2}\) because 4 is the same as \(2^{2}\).

Step 2: Equate the exponents

If \(2^{x}=2^{2}\), that tells us that \(x=2\). So our solution is \(x=2\). However, if you did not notice that 4 equals \(2^{2}\), you could also have taken the natural log or log base 2 of both sides to solve for x.

Step 3: Check the answer

Substitute \(x=2\) into the original equation to confirm the solution: \(2^{2}=4\). The left-hand side equals the right-hand side, so this confirms that the solution is correct.

Key Concepts

Exponent Rules

Exponents are rules or guidelines that help us simplify and solve problems involving powers. They appear in equations like the one given, where numbers are raised to certain powers, or exponents. Exponentiation is a shorthand for repeated multiplication. For example, in the expression \(a^b\), \(a\) is called the base, and \(b\) is the exponent. If \(b\) is a positive integer, it means \(a\) is multiplied by itself \(b\) times.

The basic rules of exponents that we often use include:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m\cdot n}\)
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
• Zero Exponent Rule: \(a^0 = 1\) if \(a eq 0\)

Understanding these rules is critical for solving exponential problems, as they allow us to manipulate the expressions into forms that are easier to work with. In the provided problem, recognizing that \(4 = 2^2\) was key to solving the equation.

Logarithms

Logarithms, often seen as the opposite of exponents, are a mathematical tool that helps us solve equations involving exponents, especially when the base or exponent isn’t obvious. The logarithm answers the question: to what exponent must we raise a certain base to get a specific number? For instance, if you see \(\log_b a = c\), it translates to saying \(b^c = a\).

Key properties of logarithms include:

• Product Rule: \(\log_b (xy) = \log_b x + \log_b y\)
• Quotient Rule: \(\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y\)
• Power Rule: \(\log_b(x^y) = y\cdot\log_b x\)
• Change of Base Formula: \(\log_b a = \frac{\log_k a}{\log_k b}\)

Logarithms simplify finding the exponent in equations where conversion directly by exponent rules isn’t feasible. In our problem, should you not immediately see base conversions, logs help in meticulously calculating the unknown exponent.

Equation Solving

Solving exponential equations means finding the unknown value in expressions gone powered up. Whether tiny and seemingly simple or large and complex, equations follow specific methods that often require aligning the bases or leveraging logarithms when the base isn't easily visible.

Equations where variables are exponents are solved by:

• Finding a common base to simplify the equation
• Using logarithms if the base isn’t easily comparable
• Isolating the variable you're solving for
• Checking the solution to ensure it satisfies the original equation

In our sample exercise, the core act was aligning the bases (\(2^x = 2^2\)), which made isolating \(x\) simple and clear, verifying that \(x=2\) is indeed the correct solution.

Base Conversion

Base conversion in exponential equations involves rewriting numbers to share the same base, as it simplifies comparison and manipulation. Finding a common base allows us to equate exponents directly, an essential step in solving the exercise solution provided: turning \(4\) into \(2^2\).

Here is what to consider:

• Identify possible bases that two numbers can share
• Convert numbers to the identified base, showing them as powers of the base
• Simplify the equation, directly equating exponents once a common base is established

Base conversion not only simplifies solving but also highlights the interconnected nature of numbers and powers. In our problem, quickly recognizing \(4\) as \(2^2\) allowed for a seamless solving process by equating the exponents, hence \(x=2\).