Problem 16 - Precalculus with Calculus Previews

Question

Solve the given exponential equation. $$ \left(\frac{1}{3}\right)^{x}=9^{1-2 x} $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{2}{3}\).

Step 1: Express with the same bases

First, we want to express both sides of the equation using the same base. The left side in the equation is \(\left(\frac{1}{3}\right)^x\), which can be rewritten as \((3^{-1})^x\). The right side \(9^{1-2x}\) can be rewritten using the base 3 as well. Since 9 equals \(3^2\), this gives us \((3^2)^{1-2x}\).

Step 2: Simplify exponents

Let's now simplify the expressions by combining the exponents using the power of a power property, which states \((a^m)^n = a^{mn}\). Thus, \((3^{-1})^x = 3^{-x}\) and \((3^2)^{1-2x} = 3^{2(1-2x)} = 3^{2 - 4x}\).

Step 3: Set the exponents equal

Now that both sides are expressed with the same base, we can set the exponents equal to each other. This gives the equation \(-x = 2 - 4x\).

Step 4: Solve the linear equation

To solve \(-x = 2 - 4x\), add \(4x\) to both sides to get \(3x = 2\). Then divide each side by 3 to isolate \(x\). So, \(x = \frac{2}{3}\).

Key Concepts

Exponential Functions

An exponential function is a mathematical expression in which a constant base is raised to the power of a variable. Exponential functions are written in the form \( f(x) = a^x \), where \( a \) is a positive real number called the base, and \( x \) is the exponent or power. This form is useful because it represents growth or decay processes where quantities change at rates proportional to their current amount. In our exercise, the expression \( \left(\frac{1}{3}\right)^x = 9^{1-2x} \) is an example of an exponential equation.
You often encounter exponential functions in real life, like compound interest, population growth, or radioactive decay. They're also core in helping solve exponential equations like the one in our exercise. Understanding these functions involves recognizing both the role of the base and the variable exponent.
These characteristics are what help determine how the graph of an exponential function behaves. For instance, when the base is greater than 1, the function shows exponential growth, whereas a base between 0 and 1 results in exponential decay.

Bases of Exponents

The base of an exponent is the number or expression that is being multiplied by itself a number of times determined by the exponent. In the equation \( a^n \), \( a \) is the base, while \( n \) is the exponent. The base determines the rate at which the value becomes larger or smaller depending on the value of the exponent.
In physical and natural sciences, bases of 10 are common (like in logarithms and scientific notation), as are bases of \( e \) in natural exponential functions. Understanding the base is essential because it affects the whole expression's behavior.

• If the base is greater than 1, repeated multiplication results in growth.
• If the base is between 0 and 1, it results in shrinkage or decay as seen in our left-hand expression, \( \left(\frac{1}{3}\right)^x \).

Switching to common bases, like base 3 in our exercise, makes solving exponential equations more manageable. This process involves converting expressions to a common base to utilize exponent properties effectively.

Exponent Properties

When working with exponential equations, several properties of exponents come into play. These rules simplify expressions and make it possible to solve equations more efficiently. Let's look at some of these key properties:

• Product of Powers Property: \( a^m \cdot a^n = a^{m+n} \). This property allows you to multiply expressions with the same base by adding their exponents.
• Power of a Power Property: \( (a^m)^n = a^{mn} \). This is used frequently in simplifying expressions, as we used in simplifying the left-hand side to \( 3^{-x} \).
• Power of a Product Property: \( (ab)^m = a^m \cdot b^m \).

In the given exercise, we extensively use the power of a power property to rewrite both sides of the equation with the same base. By doing this, we can compare just the exponents and solve the resulting equation, simplifying the problem-solving process. Recognizing and applying these properties correctly is critical when working with complex exponential expressions.

Solving Linear Equations

Solving linear equations often follows once you've simplified an exponential equation using common bases and its properties. A linear equation is expressed in the form \( ax + b = c \), where the highest power of the variable is 1.
After rewriting the given exponential equation from \( 3^{-x} = 3^{2 - 4x} \) to \( -x = 2 - 4x \), we transition to solving the linear equation. Solving these involves standard techniques:

• Isolate the variable on one side of the equation.
• Do this by performing inverse operations like addition, subtraction, multiplication, or division.

In our equation, adding \( 4x \) to both sides yields \( 3x = 2 \). Dividing both sides by 3 then simplifies it further to \( x = \frac{2}{3} \).
Linear equations, though simpler than exponential ones, require careful manipulation to isolate the unknown variable while keeping the equation balanced. With practice, these processes become familiar and useful for solving real-world problems.