Problem 9 - Intermediate Algebra

Question

For Problems \(1-34\), solve each equation. $$ 3^{-x}=\frac{1}{243} $$

Step-by-Step Solution

Answer

The solution is \(x = 5\).

Step 1: Recognize Powers of 3

Notice that 243 can be expressed as a power of 3. Specifically, \(243 = 3^5\). This is crucial because we want both sides of the equation to have the same base.

Step 2: Rewrite the Equation

Rewrite the equation by expressing \(243\) as a power of 3: \[ 3^{-x} = \frac{1}{3^5} \]

Step 3: Apply Properties of Exponents

Recognize that \(\frac{1}{3^5}\) can be rewritten using negative exponents as \(3^{-5}\): \[ 3^{-x} = 3^{-5} \]

Step 4: Equate Exponents

Since the bases are now the same (both are 3), we can set the exponents equal to each other: \[ -x = -5 \]

Step 5: Solve for x

Solve for \(x\) by multiplying both sides of the equation by -1: \[ x = 5 \]

Key Concepts

Properties of Exponents

Understanding the properties of exponents is key when dealing with exponential equations. These properties simplify complex problems, making them more approachable. Here are some fundamental properties to keep in mind:

• Product of Powers Property: When you multiply two exponents with the same base, you can add the exponents. For instance, if you have \(a^m \times a^n\), the result is \(a^{m+n}\).
• Quotient of Powers Property: When dividing exponents with the same base, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power Property: When an exponent is raised to another exponent, you multiply the exponents. For example, \((a^m)^n = a^{m \times n}\).

Generally speaking, understanding these properties helps in manipulating and solving equations efficiently. In our exercise, recognizing that \(243\) can be written as \(3^5\) and converting fractions using exponent properties was a crucial step.

Negative Exponents

Negative exponents can seem tricky at first, but they have a simple meaning. A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent.

• If you have \(a^{-n}\), it is equivalent to \(\frac{1}{a^n}\).
• Similarly, \(\frac{1}{a^{-n}}\) rewrite to \(a^n\).

In the given problem, we see the negative exponent property used to transform \(\frac{1}{3^5}\) into \(3^{-5}\). This transformation simplifies the equation solving process, as you can now compare exponents directly. Understanding this concept can greatly simplify your work with exponential functions, as negative exponents are quite common in solving algebraic equations.

Solving Exponential Equations

Solving exponential equations often involves expressing both sides of the equation with the same base. This allows the direct comparison of exponents, which simplifies the solving process.

• The first step is often identifying a common base. For instance, recognizing that \(243\) is \(3^5\) allowed us to easily work with the left side equation base \(3^{-x}\).
• Once both sides have the same base, equate the exponents. This is possible because if \(a^m = a^n\), then \(m = n\).
• Finally, solve the resulting equation for the unknown variable.

In our example, by writing both sides using the base 3, we simplified the equation, setting \(-x = -5\), and quickly solved it by isolating \(x\). This process can be applied broadly to various exponential equations, providing a powerful method to tackle algebraic problems.