Problem 8 - Intermediate Algebra

Question

For Problems \(1-34\), solve each equation. $$ 4^{x-3}=16 $$

Step-by-Step Solution

Answer

The solution is \(x = 5\).

Step 1: Recognize the Base

The equation is given as \(4^{x-3} = 16\). Notice that 16 can be rewritten as a power of 4: \(16 = 4^2\).

Step 2: Set Exponents Equal

Since the bases on both sides are the same (both are base 4), we can set the exponents equal to each other: \(x-3 = 2\).

Step 3: Solve for x

To solve for \(x\), add 3 to both sides of the equation: \(x-3 + 3 = 2 + 3\), simplifying to \(x = 5\).

Key Concepts

Solving Exponential Equations

When we tackle exponential equations, the goal is to find the value of the variable, usually denoted as "x," that makes the equation true. Exponential equations often involve variables in the exponent position, which can seem tricky at first. However, once we understand the process, it gets easier.
In the given exercise, the equation was \(4^{x-3} = 16\). The first step in solving any exponential equation like this is to express all the terms with the same base if possible. This means looking for ways to rewrite numbers on both sides of the equation in terms of powers of the same base. This makes it possible to isolate the variable.
Here are the steps in brief:

• Identify or transform each number in the equation to have the same base.
• Once the bases are the same, set the exponents equal to each other and solve the resulting simpler equation.
• Solve for the unknown variable by basic algebraic methods (like adding or subtracting on both sides of an equation).

This method turns what initially seems complex, into an easier problem that requires only basic arithmetic skills.

Bases of Exponents

In mathematics, the "base" of an exponent refers to the number that is multiplied by itself a certain number of times, which is indicated by the exponent (or power). It is a crucial concept in working with exponential equations.
For example, in \(4^{x-3} = 16\), the number 4 is the base. When dealing with exponential equations, it's often necessary to compare `bases` from both sides of the equation. In our solved example, recognizing that 16 can also be expressed as \(4^2\) enabled us to have consistent bases and rewrite the equation in useful terms.
Key points to remember are:

• If the bases of the exponents are the same, we can set the exponents equal to each other and solve the resulting equation.
• Handling consistent bases simplifies complex equations, making the solution approachable.
• Understanding how to manipulate and match bases is essential for solving a wide class of exponential equations.

This ability to adjust how we see numbers and transform them into compatible bases is foundational in mathematics.

Exponent Laws

Exponent laws provide a framework for working with exponents effectively and making exponential equations more manageable. These rules dictate how to simplify expressions and solve equations involving powers. Understanding these rules will make the process of dealing with exponents straightforward and intuitive.
Here are some key exponent laws to remember:

• The product of powers law: To multiply powers with the same base, add their exponents. For example, \(a^m \times a^n = a^{m+n}\).
• The power of a power law: To raise a power to another power, multiply the exponents. In other words, \((a^m)^n = a^{mn}\).
• The power of a product law: When a product is raised to a power, apply the power to each factor in the product separately, i.e., \((ab)^n = a^n \times b^n\).

In our exercise, recognizing that \(16\) is \(4^2\) is an application of understanding bases along with the ability to simplify using the recognition of a common base in facilitating the solution of an exponent equation.
Mastery of these laws will allow you to tackle both basic and complex exponent problems with confidence.